Thesis submitted for the degree of Master of science in nuclear physics

Shell evolution towards ⁷⁸ Ni:

Spectroscopy of ⁷⁴ Cu and ⁷⁶ Cu

Line Gaard Pedersen

Thesis submitted for the degree of Master of science in nuclear physics

60 credits

Physics Department

Faculty of mathematics and natural sciences

UNIVERSITY OF OSLO

Shell evolution towards ⁷⁸ Ni:

Spectroscopy of ⁷⁴ Cu and

76 Cu

Line Gaard Pedersen

c 2019 Line Gaard Pedersen

Shell evolution towards⁷⁸Ni: Spectroscopy of⁷⁴Cu and⁷⁶Cu http://www.duo.uio.no/

Printed: Reprosentralen, University of Oslo

Abstract

A spectroscopic study of ⁷⁴Cu and ⁷⁶Cu is presented. The work done in this thesis provides the first information on excited states in odd-odd Cu isotopes near doubly magic ⁷⁸Ni. The results can be used to tune future shell model calculations in order to predict shell configurations for more neutron-rich nuclei beyond⁷⁸Ni.

74Cu and ⁷⁶Cu have been analyzed from β-decay of respectively ⁷⁴Ni and

76Ni, created by in-flight fission of a²³⁸U beam on a⁹Be target. The experiment was performed at RIKEN Nishina Center in Japan. A total of 12 excited states in ⁷⁴Cu and 24 excited states in⁷⁶Cu have been observed. Tentative spin and parity values have been assigned to all states in⁷⁴Cu and to eight states in⁷⁶Cu based on γ-decay branching ratios, logf tvalues and systematics.

Numeric shell model calculations have been performed for⁷⁰Cu,⁷²Cu,⁷⁴Cu and⁷⁶Cu in order to compare calculations to experimental results. Combining shell model calculations and experimental results of the current thesis, the 2p_3/2 and 1f_5/2 proton orbitals have been found to cross around⁷⁴Cu.

Acknowledgements

I want to thank my supervisors Andreas G¨orgen and Eda S¸ahin for all their support throughout this project. Thank you for all the hours you have spent with me and all your input and guidance, but maybe most of all for believing in me.

I also want to thank Frank Leonel Bello Garrote for introducing me to this particular field of nuclear physics and for sharing your knowledge about the analysis of these nuclei. Lastly, thank you Jørgen Eriksson Midtbø, for being my go-to KSHELL wizard and for helping me run the many big calculations.

Thank you to everyone who participated at the EURICA experiments and pro- vided me with such great data to analyze. It really has been a lot of fun!

Is this how you make extra space in LaTeX?

Thank you to my family, Pia, Eli, Frode, Stella, and Luna, for all the good times (and food!). Thank you, Magnus, for your support and for reminding me to also relax sometimes. I look forward to having more time to spend with all of you now that this is finished.

Finally, thank you to all the fantastic people at the Nuclear Physics group, Lillefy, Stjernekjelleren and other friends I have met here, for making Blindern feel like my second home. These past years have really flown by thanks to you!

Maybe not, but it works!

I could not have written this thesis had it not been for the state of Norway providing us with excellent free education. Please let us keep it this way!

Contents

1 Introduction 4

1.1 The atomic nucleus . . . 5

1.2 A word on the nuclear shell model . . . 6

1.2.1 Why is Cu so special? . . . 8

1.3 Nucleosynthesis . . . 8

1.4 The experiment . . . 9

2 Theory 10 2.1 Beta decay . . . 10

2.1.1 Energy release inβ-decay . . . 11

2.1.2 Comparative half-life . . . 11

2.1.3 Allowed and forbidden decays . . . 12

2.2 Gamma decay . . . 13

2.3 The shell model . . . 16

2.3.1 Odd-odd nuclei . . . 17

2.3.2 Shell model calculations . . . 19

2.3.3 Isospin . . . 19

3 Experimental setup and data analysis 20 3.1 Experimental setup . . . 20

3.1.1 Radioactive Isotope Beam Factory . . . 20

3.2 Technical aspects and data interpretation . . . 25

3.2.1 Electronics and data sorting . . . 25

3.2.2 Ion-β correlation . . . 25

3.2.3 Detecting photons and how photons interact with matter 27 3.2.4 Addback . . . 27

3.2.5 Calculatingγ-ray intensities . . . 28

3.2.6 Coincidence analysis . . . 28

3.3 Data analysis . . . 30

3.3.1 Reproducing the half-lives of⁷⁶Ni and⁷⁴Ni . . . 30

3.3.2 Energy-time matrices and looking for isomeric states . . . 37

3.4 Building the level schemes . . . 40

3.4.1 A word on contaminations . . . 40

3.4.2 Identifyingγ-lines in the single spectra . . . 42

3.4.3 Studyingγγ-coincidence spectra to obtain the level schemes 42 3.4.4 Obtaining theβ-feeding and logf tvalues . . . 56

3.4.5 Spin and parity assignments . . . 56

3.4.6 Shell model calculations using KSHELL . . . 60

4 Discussion 61

4.1 Spin assignments of⁷⁴Cu . . . 61

4.2 Spin assignments for⁷⁶Cu . . . 63

4.3 State evolution of the odd-odd Cu isotopes . . . 63

4.4 Shell model calculations . . . 64

4.4.1 Calculations of⁷⁴Cu and⁷⁶Cu . . . 65

4.4.2 Mixing of states . . . 68

5 Conclusion 73 5.1 Outlook and further work . . . 74

Appendices 75 A Detailed shell model calculations 76 A.1 ⁷⁰Cu . . . 78

A.1.1 Summary file . . . 78

A.1.2 Log-file . . . 80

A.2 ⁷²Cu . . . 88

A.2.1 Summary file . . . 88

A.2.2 Log-files . . . 90

A.3 ⁷⁴Cu . . . 98

A.3.1 Summary file . . . 98

A.3.2 Log-files . . . 100

A.4 ⁷⁶Cu . . . 108

A.4.1 Summary file . . . 108

A.4.2 Log-files . . . 110

Chapter 1

Introduction

Ever since the dawn of man, humans have asked themselves where we and everything around us come from. What we are made of and how our components are made has been a question for scientists and philosophers for centuries. Today we know that everything we see is made up of atoms and we know much about their composition. A lot has been discovered about the structure of atoms, and their inner core called the nucleus. However, there is not a single theory that can fully explain the interaction between the protons and neutrons inside the atomic nucleus. The theoretical models we have must be tested by comparing predictions to experimental data, and new experimental data is needed to refine the models.

One of the most successful theories used to describe the structure of nuclei is called the nuclear shell model [1]. Still, as we will see, it is not a perfect model and we need to study experimental data in order to be able to use it for nuclei far from stability. A particularly interesting topic in that respect is the area around very neutron-rich⁷⁸Ni, and by studying it we can learn more about the shell model [2, 3]. ⁷⁸Ni is expected to be doubly magic, making it particularly interesting with regard to the shell model. In order to understand the shell model, it is crucial to investigate neighboring nuclei, as well as⁷⁸Ni itself [4]. The goal of this thesis is to obtain the level schemes of neutron-rich⁷⁴Cu and⁷⁶Cu and use these to better understand the proton-neutron interaction for odd-odd nuclei close to⁷⁸Ni. Cu has only one proton more than Ni, and⁷⁶Cu and⁷⁴Cu have respectively three and five neutrons less than ⁷⁸Ni. The shell evolution of Cu isotopes can thus tell us much about the shell evolution of Ni isotopes, which is of particular interest. This thesis presents the first investigations of the structures of⁷⁴Cu and⁷⁶Cu. No properties of these nuclei were previously known, other than their half-lives [5] and ground state spin and parity [6]. It can also be mentioned that no other odd-odd Cu isotopes with a mass number bigger than 72 have been studied to obtain their level schemes, even though the proton-neutron interaction is important to study in order to tune the shell model. Only when this is done, we can predict structures of even more unstable nuclei.

Before we dive into the analysis, let us start at the beginning.

Figure 1.1: Chart of nuclides in one of the many common ways to present it. The colors correspond to the main decay modes of the specific nucleus. All nuclei discussed in detail in this thesis are neutron-rich and decay byβ⁻decay (pink).

Numbers framed in this figure (8,20,28,...) are the so-called magic numbers, representing the shell closures for the specific number of protons or neutrons in the shell model. The figure is reproduced from [7].

1.1 The atomic nucleus

Everything around us is made of atoms, and these atoms are again made up of electrons swirling around a nucleus consisting of protons and neutrons. The numbers of protons, neutrons, and electrons give an atom its specific chemical and physical properties. For a nuclear physicist, it is the nucleus that is studied.

The familiar periodic table lists all known elements (atoms with a given number of protons,Z) in a structured manner. In nuclear physics, it is just as important to know the number of neutrons in a nucleus. Known isotopes of elements (given by the number of neutrons,N) are organized in the chart of nuclides, as illustrated in Figure 1.1.

Neutrons and protons stick together in a nucleus because they gain energy by doing so. From Einstein’s well-known formula E =mc², it is known that mass and energy are two sides of the same coin. This means that if a nucleus can lose some of its mass, it will release energy. The energy difference between Z free protons andN free neutrons, and a nucleus with proton numberZ and neutron number N, is called the binding energy. Mathematically, the binding energy of a nucleus is given as

m(N, Z) = 1

c²E(N, Z) =N mn+Zmp− 1

c²B(N, Z) (1.1)

wheremn,mp are the masses of free neutrons and protons respectively,cis the

Figure 1.2: Binding energy per nucleon,B/A, over mass number, A. The plot shows how elements lighter than iron can gain energy through fusion while the heavier elements will need excess energy to fuse. Figure taken from Ref. [8].

constant speed of light in vacuum andB is the binding energy.

If the binding energy per nucleon is plotted as a function of total nucleon number,A=N+Z, the resulting plot is as presented in Figure 1.2. The figure illustrates that the binding energy is dependent on mass. It also shows that light nuclei can gain energy by fusing together (fusion) and heavy nuclei can gain energy by splitting (fission). At the maximum of the B/A plot, we find

56Fe. As elements lighter than⁵⁶Fe can be created in stars through fusion, an interesting question here is then how elements heavier than iron are created.

Although the main focus in this thesis is not to answer this question, it can be mentioned that more information about the structure of neutron-rich nuclei might be used in various reaction codes for simulating processes where heavy elements are created.

1.2 A word on the nuclear shell model

As can be seen from the plot in Figure 1.2, the evolution of the binding energy is jagged and not as smooth as one might expect. The reason for this comes from the inherent shell structure of the nucleus. Inspired by the atomic shell model, where atomic electrons only have discrete energies, and each energy level can only include a given number of electrons, the nuclear shell model was developed in a similar way.

Scientists discovered in the mid-1900s that for some specific numbers of protons and neutrons, especially much energy was needed to remove a proton or neutron. These nuclei are thus more tightly bound than neighboring nuclei.

Because of their specialty, these numbers were named the ”magic” numbers and

Figure 1.3: Schematic representation of the nuclear shell model before (left) and after (right) introducing the spin-orbit coupling. Each energy level is identified by a set of quantum numbers, |N lji, where N is the main oscillator quantum number,l the angular momentum (assigned asl= (s, p, d, f, g) = (0,1,2,3,4)) and j the total spin, j = l±s, for each nucleon. The numbers to the right indicate how many particles can be filled in each level. In the shell model, neutrons and protons fill the different levels separately. Image modified from Ref. [12].

they are 2, 8, 20, 28, 50, 82 and 126 [9]. The magic numbers are marked in the chart of nuclides (Figure 1.1).

It was understood that the magic numbers corresponded to an energy level in the nucleus being completely filled up, and that there is a large gap between this and the next energy level. However, the physics behind how these gaps appeared where they do was not realized at first. Experimenting with harmonic oscillator potentials for the protons and neutrons and with more realistic potentials, for example by combining it with a finite square well potential, only resulted in reproducing the first three magic numbers. Introducing the angular momentum term l² to the potential helped, but it was not before Maria Goeppert-Mayer introduced the spin-orbit coupling [10] to the strong nuclear force that all the known magic numbers were reproduced [11]. Including the spin-orbit coupling gives the shell model presented in Figure 1.3.

The nuclear shell model was developed at a time where data on exotic nuclei, i.e. man-made unstable nuclei, was scarce. A result of this is that the shell model and its magic numbers are well understood for stable nuclei. How the magic numbers evolve when moving away from stability is a more recent question.

Indeed, it is shown that the magic numbers gain new features when moving

away from stability [3]. It is for example shown that the magic numberN = 20 disappears at³²Mg [13],N = 28 disappears at⁴²Si [14] and thatN = 34 appears for⁵⁴Ca [15].

1.2.1 Why is Cu so special?

When a nucleus consists of a magic number of protons and a magic number of neutrons, the nucleus is called doubly magic. It is close to these doubly magic nuclei that it is best to study the shell model. This is because the structure of nuclei is well described by relatively simple single-particle excitations, and hence also shell model configurations, near closed shells. Further away from closed shells, collective phenomena and deformations are more prominent. The most exotic doubly magic nuclei ever created is⁷⁸Ni, which has 28 protons and 50 neutrons. To what extent⁷⁸Ni really has doubly magic properties will give us an answer to how the shells change when moving away from stability. Much work has been done in this region (see for example Refs. [5, 16–24]) and the results included in this thesis will contribute to what we know about⁷⁸Ni and the region around it.

The nuclei studied for this thesis are⁷⁶Cu and ⁷⁴Cu. Having one proton more and respectively three and five neutrons less than⁷⁸Ni, they are optimal cases for studying nuclei near⁷⁸Ni. Cu isotopes have 29 protons, meaning that they are one extra proton away from magicity. Adding neutrons to Cu is thus an excellent way of studying the evolution of theZ= 28 shell gap. To this day, there have been very few studies of the odd-odd neutron-rich Cu isotopes, even though the proton-neutron interaction in this area is crucial for understanding how the shell model evolves for neutron-rich nuclei. Because the shell model needs to be tuned to experimental data, analyzing these nuclei is important. By contributing with experimental data in unstable regions of the chart of nuclei, shell predictions on even more unstable nuclei can be made.

One of the main aims of this thesis is to contribute to a better understanding of the shell model around⁷⁸Ni and thus also a better understanding of the shell model in general. The structure of⁷⁸Ni and its neighboring nuclei is important to understand in order to give any predictions about the structure of more neutron-rich nuclei surpassing ⁷⁸Ni.

1.3 Nucleosynthesis

Understanding the nuclear structure is a goal in itself, but a good understanding can also help us explain other phenomena and give a better understanding of various processes. An example of what the shell structure can help us explain is how elements heavier than iron are made in the universe. Although this is not a main focus in the thesis, it can be mentioned that improved shell model calculations will indirectly help us better understand the processes in which heavy elements are made in the universe.

On Earth,⁷⁸Ni and its surrounding nuclei have to be created in laboratories using modern equipment. In the universe, it can be created under specific circumstances as a part of the nucleosynthesis for heavy elements. To explain this, we have to take a look at how elements are made in the universe.

Stars get their energy by fusing hydrogen into helium. The most massive stars can fuse heavier elements towards the end of their life and create elements as heavy as iron. As mentioned earlier, fusion beyond iron requires energy rather than releasing it, so the fusion process stops here. The creation of heavier elements must thus happen in other ways.

There are several theories of how and where the creation of heavy elements takes place in the universe. One of the most discussed processes is the rapid neutron-capture process (or r-process) [25], in which nuclei under extreme pres- sure can capture many neutrons quickly. For this to happen, there must be an environment where the pressure and neutron density are high enough for the neutrons to be captured by a nucleus faster than the newly created nucleus can β-decay. One candidate for where these conditions are satisfied is in neutron- star collisions. This hypothesis was strengthened after the gravitational waves of two neutron stars were detected in 2017 [26].

One process that can be responsible for the isotope abundance observed in our solar system is the so-called weak r-process. This primarily forms the A ∼ 80 abundance peak, and is associated with neutrino winds from core- collapse supernovae [27]. Knowing the neutron capture rates for nuclei in the areas of A∼80 is important in order to simulate the weak r-process. From a sensitivity study by Surman et. al. [28] it was concluded that both ⁷⁶Cu and

74Cu are among key nuclei in determining the path of the weak r-process. Their neutron-capture cross-sections are therefore important, but not yet possible to measure in laboratories. Instead, I hope that the level schemes I obtain in this thesis may guide various simulations that can use the level distribution to obtain the neutron-capture cross sections.

1.4 The experiment

The data analyzed in this thesis comes from a big collaborative experiment performed at RIKEN Nishina Centre, Japan, in 2012. The main goal of this experiment was to study nuclei in the region of⁷⁸Ni and learn more about their structure and half-lives. This is both to provide experimental data for tuning shell model parameters and to provide data for astrophysical calculations. Half- lives of unstable nuclei are essential for nuclear astrophysics calculations. From this experiment, half-lives of 20 neutron-rich nuclei were measured, five of which for the first time, as presented in Refs. [5, 29]. The data has also been used for spectroscopic studies, as in Refs. [30, 31].

The same data could also be used to study structures of nuclei previously not studied. Data for the various nuclei were distributed among the collaborators of the experiment, and the Cu data were given to the Oslo group. The odd-even nuclei⁷⁵Cu and⁷⁷Cu were analyzed for F. L. Bello Garrote’s Ph.D. thesis (Ref.

[32]). While odd-even Cu isotopes are good for studying single-particle excita- tions, studying odd-odd Cu isotopes will contribute to a better understanding of the proton-neutron interaction. This is what I will do by obtaining and studying the levels schemes of⁷⁴Cu and⁷⁶Cu.

Chapter 2

Theory

The goal of this chapter is to present the theory relevant to the further discus- sions in this thesis. I will go into three topics, namelyβ-decay,γ-decay and the shell model. Analysis of the nuclei in the next chapters will be done by studying

γ-decay following fromβ-decay, and it is important to have a good understand-

ing of these types of decay before we discuss them in the analysis. Towards the end of this thesis, my results will be compared to numerical calculations of the shell model to determine how well our theories fit with experimental data.

Therefore, I want to go through the shell model and the theory behind it here, to ease later discussion about the shell model.

2.1 Beta decay

One of the ways nuclei can decay is through beta decay (or β-decay). We separate the two processes of β⁺ and β⁻. In both processes the nucleus will decay into a neighboring nucleus of the same mass number (an isobar), either by turning a proton into a neutron (β⁺) or a neutron into a proton (β⁻). These processes are three-body problems, as a positron/electron and a neutrino/an- tineutrino are emitted from the nucleus as a part of the decay. We can write the reactions as:

β⁺: ^A_ZXN →^AZ−1YN+1+e⁺+ν

β⁻: ^A_ZX_N →^AZ+1Y_N−1+e⁻+ν

If a nucleus is surrounded by electrons, there is also an alternative to β⁺- decay. This is called electron capture, and it happens when an electron has an overlapping wave function with the nucleus. The electron can be absorbed by the nucleus, making the electron and a proton convert into a neutron-neutrino pair. Schematically, this can be described as:

EC:^A_ZXN+e⁻ →^AZ−1YN+1+ν

The decay relevant for this thesis is β⁻-decay. Whenever β-decay is men- tioned later in this thesis, it is always referred to β⁻-decay.

When a nucleusβ-decays, it seldom leaves the daughter nucleus in its ground state, but rather creates the daughter in an excited state. This can be exploited

in order to study the structure of the daughter, which canγ-decay to lower-lying states. The energy of each emittedγ-ray corresponds to the difference in energy of the nucleus before and after theγ-decay. It is exactly this process ofγ-decay following from β-decay that is used to obtain the results in this thesis. The

β-decay of⁷⁶Ni and⁷⁴Ni is used to measureγ-decay of the daughters,⁷⁶Cu and

74Cu respectively, and use thisγ-decay information to build their level schemes.

2.1.1 Energy release in β-decay

For β-decay, a three-body problem, the released energy will be distributed among the emitted electron, antineutrino and the nucleus. The energy released in a decay-process is referred to as theQ-value, and is defined as the difference between the initial and final nuclear masses:

Qβ=

mN(^A_ZX)−mN(^A_Z−1Y)−me

c², (2.1)

where mN(^A_ZX) andmN(^A_Z−1Y) are the masses of the nucleus before and after the decay, and emis the mass of the emitted electron. The neutrino mass can be ignored here, as its mass is several orders of magnitude smaller than the electron mass [33].

The Qβ-value, or energy released, is distributed among the electron and antineutrino. The recoil energy of the nucleus is small and can be ignored. In experiments, it is most feasible to measure the electron energy, which will give a continuous function. The Q_β-value will then be the maximum kinetic energy that can be measured for the electron, corresponding to the electron receiving all the energy and the neutrino none.

2.1.2 Comparative half-life

It is not always possible to measure the spin and parity of states in the daughter nucleus directly, so other methods have to be applied. In this section, it will be described how transition rates can be used to obtain spin and parity information.

The following is all gotten from Fermi’s theory ofβ-decay. A general result for transition rates is the probability for decay described by Fermi’s golden rule:

λ= 2π

¯

h|Vf i|²ρ(Ef), (2.2)

where ρ(E_f) is the density of final states. V_{f i} is the interaction between the initial and final state and is defined as:

Vf i=g Z

ψ_f^∗Vβψidv=gMf i, (2.3) where the constant g determines the strength of the decay. Mf i is the matrix element which gives the transition rate from an initial state i to a final state f, and ψi and ψf are the the wave functions of the initial and final systems (including the wave function of the electron and anti-neutrino) and Vβ is the operator responsible forβ-decay.

As mentioned earlier, this all applies to only the Fermi theory ofβ-decay. To extend the Fermi theory ofβ-decay to also include Gamov-Teller decay (which will be presented in the next section), we can split the matrix element and

strength constant in two: one for Fermi decay and one for Gamov-Teller. We write this as:

|Vf i|²=g_F²M_F²+g_GT² M_GT² . (2.4) Further calculations and proofs are given in Chapter 9.2 in Ref. [34]. Here I only include the results, where we find the decay rate, λ, to be given as:

λ= ln 2

t_1/2

= g²|Mf i|² 2π³¯h⁷c³

Z p_max

0

F(ZD, pe)p²(Qβ−Te)²dpe

= g²|Mf i|²m²_ec⁴

2π³¯h⁷ f(ZD, Qβ) (2.5)

whereF(ZD, pe) is the Fermi function, which describes the Coulomb interaction between the nucleus and the electron, andf(ZD, Q) is called the Fermi integral.

From this, we see that the decay rate is given by the Fermi integral, the matrix element and a series of constants. To find the transition rates is thus to calculate thef tvalues, which are inversely proportional to the square of the matrix elementMf ifor a particular transition:

f t_1/2∼ 1

|Mf i|². (2.6)

As the partial half-lives take a wide range of values, it is common to use the logf t value, also known as the comparative half-life. Figure 2.1 shows known logf t values for decays to states with known spin and parity.

Even if the logf tvalues of each state is known, it does not necessarily give an unambiguous classification of the decay. This is because the various types of β-decay introduced in the next section have some overlap with each other. This is more thoroughly discussed in Ref. [35] and shown in Figure 2.1. The logf t values can still be used as guidelines when determining spin and parity of the daughter nucleus excited states, or at least to constrain possible spin values in the daughter nucleus.

2.1.3 Allowed and forbidden decays

The probability of β-decay to a particular state in the daughter nucleus is de- scribed by the matrix elements,Mf i, as well as the spin and parity of the final state, compared to the initial state of the mother. The termsallowed andfor- bidden decays are used to describe how probable specific transitions are. Note that a forbidden decay is less likely than an allowed transition, but not impos- sible. It is also important to remember that the total angular momentum must be conserved in theβ-decay process.

One way to define types of β-decay, is by the angular momentum carried away by the total spin of the electron-antineutrino pair. The first case is called Fermi transitions and defines transitions where the emitted electron and an- tineutrino couple to a total spin S = 0, giving a total angular momentum change ∆J = 0. The second case is the Gamow-Teller transitions, where the electron-antineutrino pair couple to S = 1, giving the angular momen- tum change ∆J = 0,±1. In order to carry away more angular momentum, the electron-antineutrino pair needs to carry orbital angular momentum.

Table 2.1: Types of β-decay and how allowed they are, or how likely they are to occur. Note that the selection rules often can be satisfied by both Fermi and Gamov-Teller dacays, leading to a mixture of the two.

Forbiddenness ∆J ∆π

Superallowed 0 no

Allowed 0, 1 no

First forbidden 0, 1, 2 yes Second forbidden 1, 2, 3 no Third forbidden 2, 3, 4 yes

The total angular momentum of the emmited particles consists of the angular momentum (L) and the total spin (S). Allowed transitions are characterized by L = 0, while everything with L >0 is characterized as forbidden transitions.

Selection rules for change in total spin of the nucleus during β-decay are given as:

∆J =L−1, L, L+ 1

∆π= (−1)^L

where ∆π= 1 corresponds to no parity change and ∆π =−1 corresponds to parity change.

From all this, we can list the various decays in Table 2.1. The same clas- sifications and rules apply for β⁺-decay as well, when we look at the emitted positron-neutrino pair.

To sum up, a β-decay transition is described by how likely it is to occur within a specific time, given by the change of spin and parity from the mother to the daughter nucleus. Before we move on toγ-decay, one last thing we need to define is β-delayed neutrons, which we might see in experiments like the ones studied here. For very neutron-rich nuclei the Q_β-value is larger than the neutron separation energy of the daughter nucleus. This means that theβ-decay can populate states in the daughter above the neutron separation energy and followingγ-decay from the daughter will compete with neutron emission. These emitted neutrons are referred to asβ-delayed neutrons.

2.2 Gamma decay

At low energies, the most common way an excited nucleus gets rid of excess energy is through gamma decay (orγ-decay). In this process, a photon carrying the energy difference between the initial (Ei) and final (Ef) states,E=Ei−Ef, is emitted. Like in β-decay, the probability of γ-decay within a certain time is also connected to the spin and parity of the final and initial states. We can exploit this when determining spin and parities in the various states in a nucleus.

From the γ-ray spectra, we see how strong the different transitions are based on how big their peaks are.

We categorizeγ-decay into the type of decay (electric or magnetic) and mul- tipolarity. If nuclei have spin alignment, the type and multipolarity can be measured by studying the angular distribution of emittedγ-rays. In the experi- ments analyzed in this thesis, the spins of the nuclei were randomly oriented. It is possible to studyγ-ray cascades and find the relative distribution between the

Figure 2.1: Distribution of the different allowed and fobidden decays over their obtained logf t value, as found in Ref. [35]. This illustrates that even when a state’s logf tvalue is known, it is not trivial to assign its spin and parity values.

Table 2.2: Table of γ-decay selection rules. Lowest possible multipolarity is always preferred, while for the same multipolarity,Eλwill always be preferred overM λ.

|∆I| 0 1 2 3 4

∆π= yes E1 E1 M2 E3 M4

(M2) (M2) E3 (M4) E5

∆π= no M1 M1 E2 M3 E4

E2 E2 (M3) E4 (M5)

differentγ-ray transitions, but this relies onγγ-coincidences and requires a high level of statistics. In our experiments, it is more feasible to study the intensities of the various transitions in the current analysis. For this, theγ-decay selection rules are considered. They are presented in Table 2.2, where we use:

|∆I|=|If−Ii|,|If−Ii|+ 1, ..., If+Ii,

whereIi andIf and the spins in units of ¯hof the initial and final state. Change in parity is given as (−1)^λ or (−1)^λ+1 for electric Eλ-transitions or magnetic M λ-transitions, respectively.

Specific transitions might require high multipolarities to decay, making the probability of these decays low compared to other γ-transitions in the same nucleus. States where this is the case are known as isomeric states. In cases of high multipolarity and longγ-decay half-life, β-decay can become competitive for unstable nuclei. For illustration, see Figure 2.2.

Figure 2.2: Illustration of an isomeric state. It is unlikely (but not necessarily impossible) toγ-decay to the ground state, giving the state a long half-life. De- caying from the isomeric state directly throughβ-decay might be more probable, given that the nucleus actually is unstable andβ-decays in the first place. Note that because of the big I^π differences, the β-feeding to the daughter nucleus states will be different from the isomeric state and ground state in the mother.

2.3 The shell model

The nuclear shell model is one of the most successful models used to describe the atomic nucleus. It was briefly presented in the introduction, but in this chapter, I will go more deeply into the theory behind it. Figure 1.3 is the simple scheme representing the shell model, made by using a slightly modified harmonic oscillator potential. In reality, the energy levels have to be calculated for each nucleus. As nuclei are many-body quantum particle systems, they are impossible to calculate numerically with today’s computers. In order to perform calculations on big-Anuclei, we must reduce the problem to something we actually can calculate. Here the shell model comes in handy.

One adjustment to reduce the required computation power is to reduce the number of particles effectively contributing to the potential. A common way to do this is to describe the nucleus as a few particles interacting with each other outside an inert doubly magic core. Thus we reduce the problem by only studying some of the nucleons. The particles outside the inert core are called valence particles, and the orbitals they can inhabit are referred to as the valence space. We usually choose the inert core and valence space by guidance from the shell model. Studying neutron-rich Cu isotopes, we see that we have more than 28 protons and neutrons respectively, making a core of ⁵⁶Ni a good guess for the inert core. For ⁷⁴Cu or⁷⁶Cu, this would give 1 proton and respectively 17 or 19 neutrons outside the core. The advantage of using the ⁵⁶Ni core is the relatively small valence space, making these calculations possible to run on a laptop. However, the valence space with only one proton is very limited, and we cannot expect such calculations to be perfect. For more accurate calculations, we can use a core of ⁴⁸Ca where we demand the bottom 20 protons and 28 neutrons to be part of this inert core. These calculations are big computational problems and require a supercomputer.

When only calculating the interaction of our valence nucleons and ignoring interactions with the nucleons in the core, we must compensate by changing the interaction between the valence particles. To do this, we introduce what we call an effective interaction and effective charges. This way, it is possible to reproduce measured values by tuning the effective interaction and charges.

One great success of the shell model is its ability to predict the spin and par- ity of different states. This is done by assessing how many uncoupled neutrons and protons there are in a given state. If there is an even number of particles in a particular state, defined by quantum numbers N lj, all particles are cou- pled and their total contribution to the spin and parity would be zero. For an odd number of nucleons, there would be one particle contributing to the spin and parity of the nucleus, while the remaining particles would couple and not contribute to the total spin and parity. Note that in terms of filling the energy levels, protons and neutrons are completely unaffected by each other, and it is essential to separate the two.

Let us start with the odd-even Cu isotopes. With an even number of neutrons and 29 protons, the total spin and parity of the ground state are given by the single unpaired proton. According to the shell model in Figure 1.3, the lowest energy state the unpaired proton can occupy is the 2p3/2state. Indeed, checking all the odd-mass Cu isotopes from stable (A= 63,65) to neutron-rich up to and including A = 73, all of them have a ground state I^π = 3/2⁻ [7]. This is a great success of the shell model. However, for the next odd Cu isotopes, namely

Figure 2.3: Figure from [21], showing how levels can be shifted when adding more neutrons. This is calculated energies for the Ni isotope chain, based on experimental results from odd-even Cu isotopes.

75Cu and ⁷⁷Cu, the ground states are found to be 5/2⁻ states [7], suggesting the uncoupled proton is in the 1f5/2level! [19, 21]

In order to discuss what happens when adding enough neutrons to send the uncoupled proton into the 1f5/2level, we need to introduce the tensor force. Up until now, we have described the proton and neutron shells of being independent of each other. In reality, however, they actually affect each other. The tensor force has been introduced analytically and as a robust general feature, for the first time by Otsuka et. al. [36], and is a force that acts between nucleons in different single-particle orbits. I will not go into the math here, but a few words about the interaction will now follow. When changing the number of nucleons of one type (e.g. neutrons) in a specific orbital, the tensor force can affect the energy of spin-orbit partner orbitals for the other type (e.g. protons). This will change the relative energies of the orbitals, and can also lead to an inversion of the orbitals. How much the tensor force pulls or pushes depends on the relative spin orientation, i.e. whetherj= +1/2 orj=−1/2. In the case of neutron-rich Cu isotopes, neutrons are added to theg_9/2orbit (withj =l+1/2) which pushes the protonf_5/2orbital (withj=l−1/2) down and pullp_3/2(withj =l−1/2) up [36]. As numerical and experimental studies of odd Cu isotopes have shown, the tensor force affects the splitting betweenf_5/2andf_7/2enough to invert the 1f5/2 and 2p3/2 levels when adding many neutrons. See Figure 2.3.

To determine how much the levels cross when moving away from stability, it is necessary to compare the model with experimental data. Data from the same experiment as the one studied in this thesis has already shown where the 1f5/2 and 2p3/2 levels shift for the odd Cu isotopes. In order to see if this also is the case for odd-odd Cu isotopes, more analysis is needed.

2.3.1 Odd-odd nuclei

When studying odd-even nuclei, the ground state spin and parity is given by the last uncoupled nucleon. In odd-odd nuclei there are two uncoupled nucleons,

Figure 2.4: Schematic filling for the lowest energy state in ⁷⁴Cu. With one uncoupled neutron (green) in p_3/2 and one uncoupled proton (red) in g_9/2, it is the combination of these two nucleons that decide the spin and parity of the ground state. Semi-magic number 40 has been added in dotted circle, as the gap between g_9/2 and p_1/2 is relatively large, and it has been shown to be a magic number for some neutron-rich nuclei. See for example Ref. [37].

one proton and one neutron. These will couple with each other and together give the ground state spin and parity. In the example of ⁷⁴Cu as presented in Figure 2.4, there is one proton in p_3/2 that couples to one neutron ing_9/2. We write this asπp_3/2⊗νg_9/2. The total spin can then take all integer values from the absolute difference of the two spins to the sum of the two spins:

I=

9 2 −3

2

, ...,9 2+3

2 = 3,4,5,6.

Had the proton instead been in f_5/2, we would have πf_5/2⊗νg_9/2 and the ground state spin could have been:

I=

9 2 −5

2

, ...,9 2+5

2 = 2,3,4,5,6,7.

This illustrates how it is much harder to predict the ground state spin of odd-odd nuclei than of odd-even nuclei.

The g_9/2 state is an intruder state, e.g. an orbital with a different major oscillator shellN than the other close-lying states. While theg_9/2state is from N = 4, the other states in Figure 2.4 are fromN = 3. As we have coupling from one positive and one negative parity state, the total parity is negative for all the spin states obtained by coupling withg_9/2.

I would again like to remind the reader that nothing was known about the structure of⁷⁴Cu and⁷⁶Cu before the analysis in this thesis was performed. In the later chapters of this thesis, I will come back to the shell model and dis- cuss the various configurations found by comparing experimental and numerical results.

2.3.2 Shell model calculations

We can describe the effective interaction between two valence nucleons by the so-calledtwo-body matrix element (TBME) as:

hφ1|Heff|φ2i, (2.7)

where φ1 and φ2 are the wave functions of two valence nucleons, and Heff is the effective interaction for a given valence space. Nuclei studied in this thesis consist of many particles and it is necessary to put some constraints on the system in order to calculate their excited states. The model space for⁷⁴Cu and

76Cu chosen to be studied in this thesis includesp_3/2, f_5/2, p_1/2 and g_9/2 for both protons and neutrons, and we use a ⁵⁶Ni (28 protons, 28 neutrons) core.

The main focus is to try to understand the interaction between the uncoupled proton and an uncoupled neutron, and the TBME must be fitted to experimental data. As the information about odd-odd nuclei in this neutron-rich region is scarce, results from this thesis will help us better understand the proton-neutron interaction.

Various interaction codes can be used to perform shell model calculations.

For the calculations done in this thesis, the open-access code KSHELL [38] was used, along with the interaction code JUN45 [39]. Several other interaction codes were tested, but comparison with known experimental states of⁷⁰Cu and

72Cu showed that JUN45 gave the best results.

2.3.3 Isospin

Isospin is a concept introduced to show the similarities between protons and neutrons. If the electric force is neglected, protons and neutrons are very much alike, with almost the same mass and with the same strong nuclear force between them. It was Werner Heisenberg who first introduced isospin [40] to address exactly this.

The isospin formalism in nuclear physics suggests that protons and neutrons are the same particles with isospinT = 1/2. Like we in regular spin can orients to be either up or down, the same can be done with isospin. The only difference between protons and neutrons are then that the isospin projection along the z-axis isT_z= +1/2 for neutrons andT_z=−1/2 for protons¹.

Using the isospin formalism, shell model calculations for odd-even nuclei become easier as we end up with almost the same results for mirror nuclei (nuclei with opposite numbers of protons and neutrons of each other). The small difference can be explained by adding the Coulomb interaction again.

If we instead have an odd-odd nucleus the results can vary quite a lot. The TBME is very similar for a neutron-neutron interaction and a proton-proton interaction. These interactions always haveT_tot = 1, and are also very similar to proton-neutron interactions with T_tot = 1. However, the proton-neutron interaction can also couple to T_tot = 0, and to get information on the TBME forTtot= 0 we need to study odd-odd nuclei.

1This is opposite in particle physics, where an up quark is given Tz = +1/2 and down quark is givenTz=−1/2 giving the neutron a total ofTz=−1/2 and the protonTz= +1/2.

In nuclear physics, there is often more neutrons than protons and the formalism is chosen such that the vertical isospin part usually is positive.

Chapter 3

Experimental setup and data analysis

3.1 Experimental setup

The data for this thesis was taken as a part of the EURICA (EUROBALL RIKEN Cluster Array) project [41] which took place in RIKEN Nishina Center (RNC) in Japan, with the experiment carried out at the Radioactive Isotope Beam Factory (RIBF) at RNC [42]. The data is collected from two separate experiments performed in November and December 2012. At RIBF it is possi- ble to produce highly exotic neutron-rich nuclei using in-flight fission of a²³⁸U beam on a⁹Be target. Within the EURICA project, the EUROBALL Ge clus- ter detectors were mounted at RIBF to ensure excellent γ-ray detection with high resolution. Information about the experiment and the experimental site is presented below.

3.1.1 Radioactive Isotope Beam Factory

The various components used at the Radioactive Isotope Beam Factory are shown in Figure 3.1. For the experiments subject to this thesis, a primary beam of²³⁸U was used. The beam was accelerated up to 345 MeV/nucleon by a chain of four cyclotrons, two of which are present in the figure: the Intermediate stage Ring Cyclotron (IRC) and the Superconducting Ring Cyclotron (SRC).

The average beam intensity for both runs was around 10 pnA.

BigRIPS

After being accelerated, the primary ²³⁸U beam was pointed at a 3 mm thick

9Be target which induced fission of the²³⁸U ions. As shown in Figure 3.1, this marked the beginning of the path through the superconducting in-flight sepa- rator BigRIPS. BigRIPS is a two-part structure of 14 superconducting triplet quadrupoles (STQ) and six room-temperature dipoles, both designed to separate the various fission products. Each STQ is composed of three superconducting quadrupoles installed in a single cryostat [44]. In the present case, BigRIPS was tuned to optimize transmission of neutron-rich nuclei around ⁷⁸Ni and to sup-

Figure 3.1: Schematic figure of the Rare Isotope Beam Factory at RIKEN taken from [43].

press the less exotic unwanted nuclei. The first stage of BigRIPS was employed for selection and purification of the isotopes produced in the fission reaction by comparing magnetic rigidity (Bρ) and energy loss (∆E), in the so-called Bρ−∆E−Bρmethod. Energy loss is proportional toZ and is characteristic for each element. The magnetic rigidity is dependent on the mass to charge ratio (A/Q), where we in this experiment can useQ=Z. This is because the energy of the beam is too high for the electrons to stay attached to the nucleus and all atoms will be stripped of electrons. By measuring ∆E and Bρwe can identify exactly which nuclei are produced. In the second stage of the BigRIPS structure, time-of-flight (TOF) information is also taken into account and par- ticle identification is done by the Bρ−∆E−T OF method. A resulting plot from the particle identification process is shown in Figure 3.2.

After undergoing fission, the particle beam will experience a slight spread in momentum and angle. To correct for this, BigRIPS has many bends to allow a higher momentum resolution. The particle identification is mainly done in the second stage, as shown in Figure 3.1.

As illustrated in Figure 3.1, the ZeroDegree spectrometer is constructed as an extension to BigRIPS. This is mainly to increase the time-of-flight path to perform additional particle identification, especially important for experiments where a target is used at a focal point and the outgoing particles also must be identified. Data for several nuclei in the ⁷⁸Ni region were obtained in these experiments and good particle identification was crucial in order to study this region where so little was known about the nuclei in advance.

Decay station

After the ZeroDegree spectrometer, the secondary beam finally reached the de- cay station placed at focal plane F11. The ions were implanted in the WAS3ABi (Wide-range Active Silicon-Strip Stopper Array for Beta and ion detection) de- tector in order to study their β-decay properties. Thin aluminum foils were used to degrade the energy of the fission fragments to ensure that they were

Figure 3.2: Particle identification plot, showing how measuring the mass-to- charge ratio and the charge (corresponding to atomic number) could determine what nuclei are detected. As examples,⁷⁸Ni and⁸⁰Zn are marked together with their corresponding background. Reproduced from [29].

stopped in one of the eight layers of silicon detectors, and preferably in the middle layers. Each layer of WAS3ABi is designed as 16 strips vertical and 16 strips horizontal double-sided silicon-strip detectors (DSSSDs) of the type Mi- cron Model W1-1000. See Figure 3.3 for illustration and Figure 3.5 for picture.

The design makes them position sensitive and we get a grid of positions where the particles can be detected. As well as measuring the incoming particle, these detectors can also detect the electron emitted as part of the β-decay. For the data analysis, only the events where the electron was detected in the same or in a neighboring pixel as the incoming particle were interpreted as trueβ-decays.

This was chosen to discard background and random events, even though the statistics overall were slightly reduced by this decision.

12 seven-element Ge cluster detectors [45] of EUROBALL [46] were installed around the DSSSDs at focal point F11 [47]. Figure 3.4 shows a schematic illustration of the DSSSDs surrounded by Ge detectors. A photograph of the setup is presented in Figure 3.6. The detector arrangement made a total of 84 individual Ge crystals be arranged around the decay stations to form a pseudo- spherical geometry with an inner diameter of 38 cm, covering a large fraction of the solid angle. With the EUROBALL detectors the detection efficiency is high and the energy resolution is proven to be better than 3 keV at Eγ = 1.3 [43]. The absolute full peak efficiency of the 12 clusters was approximately 8%

at 1 MeV [21].

Figure 3.3: One layer of DSSSD with a 16×16 grid, making them position sensitive. Included in the illustration is the detection of a heavy ion (red dot) as well as an electron (smaller, blue dot) detected in a neighboring pixel. This implies the detection of aβ-decay event.

Figure 3.4: Eight-layered DSSSD surrounded by germanium detectors. There were in total 12 Ge detectors used at the experiment, but only three are drawn here for simplicity.

Figure 3.5: Picture of the WAS3ABi detector, taken from reference [29].

Figure 3.6: Picture of the EURICA array surrounding the DSSSDs, taken from reference [32]. The situation is schematically presented in Figure 3.4.

3.2 Technical aspects and data interpretation

3.2.1 Electronics and data sorting

A pre-sorting of the data was performed internally by the collaboration at RIKEN. The pre-sorted data was later distributed to the collaborators of the EURICA project, and the Oslo group got our set of data. A few words on the data pre-sorting will still follow.

3.2.2 Ion-β correlation

Every signal coming from the Si or Ge detectors is digitized. The energy and time information is extracted from the signals, and we get a stream of data containing the ID of the detector giving the signal, the energy of the signal, and a time stamp. The digitizers ran 100 MHz, giving a time resolution of 10 ns.

With the DSSSDs it is easy to determine the interaction point of the in- coming particle. Ions will leave the largest energy signal, while the β-decay electrons will leave a smaller energy signal. The electrons will also scatter into several pixels within the same, or even a different, DSSSD layer. In this section, I will discuss the ion-β correlation, which is the spatial correlation between the implanted Ni ion and subsequent β-electrons emitted in theβ-decay process. I will go through how these were applied to increase the number of good events while also trying to keep theγ-ray spectra as clean as possible.

There are mainly two conditions applied in the data analysis. The first con- dition is that the implanted ion and the subsequent β-electron were detected in the same DSSSD layer of WAS3ABi. Events with more DSSSD layers in- volved were in other words rejected. A ∆z variable describing the distance in z-direction, or between the DSSSD layers, was introduced and a condition for this to be zero for the inclusion of an event was set. A detailed description can be found in Ref. [29].

The second condition introduced relied on how scattering ofβ-electrons will take place in more than one pixel, due to the multiple scattering of the high- energy electrons. For this, the variable ∆xywas defined as the distance travelled in thexy-plane:

∆xy=q

(xβ−xion)²+ (yβ−yion)²

where xβ and yβ are the positions of β-electrons in xand y directions (along the detector plane) andx_ion andy_ion are the positions of the implanted ion in xand y directions. When the electrons were detected in more than one pixel, the ∆xy was obtained by using an energy weighted mean position of the pixels giving the signals in corresponding spatial direction. Detailed description can be found in References [29] and [32].

Figure 3.7 shows the number of detectedβ-events for each combination of scattered electrons obtained for⁷⁶Cu. From this figure, we see that putting the strict condition of detecting the ion and the scattered electron in the same pixel gives near zero counts. For the analysis for this thesis, ∆xy <2 was chosen as this gave the best balance between the number of counts and impact onγ-ray spectra quality.

76Ni

Entries 1792887

Mean 1.224

Std Dev 0.5956

0 0.5 1 1.5 2 2.5

0 50 100 150 200 250

103

×

76Ni

Entries 1792887

Mean 1.224

Std Dev 0.5956

beta.deltaxy {Mass==76&&beta.deltaz==0}

Δxy

Counts

Figure 3.7: Number of counts obtained for each ion-β correlation for⁷⁶Cu. For the analysis, ∆xy was set to be smaller than 2 pixels, and all possible configu- rations of accepted electron scatter within one DSSSD layer are illustrated here along with their respective number of counts detected.

3.2.3 Detecting photons and how photons interact with matter

In order to discuss the different photon detection techniques used in the exper- iment, a few words on how photons interact with the detectors will now follow.

There are three main ways photons interact with matter, and the probability of each interaction depends on the photon energy. The three main ways a photon can interact with matter are:

• Photoelectric effect (total absorption)

• Compton scattering

• Pair production

A description of each will now be presented.

Total absorption, or the photoelectric effect, is when a photon gives all its energy at once to an atom or an electron in a substance. For γ-ray detection, this is the ideal case because it is then easy to know the entire energy of the given photon. The second way a photon can interact with matter is through what we call Compton scattering. Here the photon will give some of its energy to an electron and scatter off with less energy in a new direction. Lastly, we have pair production, which consists of the photon energy being converted into an electron-positron pair. It happens in the vicinity of a nucleus, and to conserve momentum the nucleus must take some of the momentum. The likelihood of this happening is large at high energies and for heavy nuclei.

The probability distribution of these types of interaction depends heavily on photon energy. For low energies, the total absorption is most probable, while for slightly higher energies the Compton scattering effect increases in probability.

Pair production cannot happen unless the energy is higher than 1.022 MeV corresponding to the energy needed to create one electron-positron pair of mass 511 keV each.

All these effects are visible in a rawγ-spectrum where the background has not been removed. Although much background can be removed, there will still be some left in the spectra we end up studying.

3.2.4 Addback

For the analysis of the present data, it is important to know how the Ge detec- tors are built. Instead of separate detectors in an array with shielding around each detector, the EUROBALL detectors are built like seven-element clover de- tectors, meaning that there are seven separate Ge crystals put together to make one bigger detector. See Figure 3.8 for illustration. One advantage of this is that a bigger area or volume around theγ-decay events are covered with detectors.

However, it is probable that Compton scattering ofγ-rays with energy near or above 1 MeV happen between the detectors in such configurations.

There are different ways to deal with the Compton scattering between the detectors. Here we introduce the concept of addback, which has been crucial for the data analysis in this thesis. Using addback means that two counts in different neighboring detectors are put together into one event and their energies are summed. If for example Detector 1 measures aγ-photon with energyE₁and Detector 2 measures a photon with energy E₂ simultaneously, we can assume

Figure 3.8: One seven-element clover Ge detector. A photon can easily scatter between the different hexagon detectors, as illustrated, leaving a signal in two different crystals.

that it was, in fact, one photon of initial energyE =E1+E2 which scattered between the two detectors.

3.2.5 Calculating γ-ray intensities

To study how strong the different transitions are, the EURICA detection effi- ciency had to be determined as a function of γ-ray energy, and used to correct

theγ-ray intensities. This had to be done separately for the two experiments.

The standardγ-ray procedure consisted of performing Gaussian fits to allγ-lines identified as transitions in the studied nucleus. We used the Radware analy- sis program [48] for this, and the obtained areas of each peak would later be converted to also account for the efficiency function.

The efficiency function was determined prior to the present thesis, using the ¹⁵²Eu and ¹³³Ba γ-ray sources with known transitions, covering energy values from 80 keV to 1408 keV. The function obeys the form of a widely used empirical formula given in Ref. [48] and more details about the the efficiency calibration procedure can be found in Ref. [29]. For the uncertainty analysis

of γ-ray intensities, a script by the author of Ref. [32] was used. The script

included the efficiency function of the Ge detectors with calibration errors and took the Gaussian fits with uncertainties as inputs. This way, we could ensure that the obtained intensities had uncertainties where both the fit of peaks and the intensity function was taken into account.

3.2.6 Coincidence analysis

Physically speaking, our decay sequence goes as an incidentalβ-decay followed

byγ-decays, one after the other. Experimentally speaking, however, everything

happens quickly as though it all happens simultaneously. When events are detected this close in time, we say that they are in coincidence with each other.

The data is stored in a way that it is possible to see which events happened in coincidence. By using the coincidence relations ofγ-rays, we can build the level schemes for the studied nuclei.

When building a level scheme, we apply a coincidence gating. This is done by selecting a small range of energy channels corresponding to a full peak, and see what other γ-ray energies were detected in coincidence with the selected channels. In order to do this, however, it is first necessary to create a so-called

(a) A simpleγγ-matrix. (b) Obtained level scheme.

Figure 3.9: Simple example of how to interpret a γγ-matrix (Figure (a)) and use it to obtain the level scheme. The dashed lines indicate that we gate on energyE1and see energiesE2andE3in coincidence. In (b) we see the obtained level scheme from the simple coincidence analysis done on theγγ-matrix.

γγ-matrix.

An example of a simple γγ-matrix is shown in Figure 3.9a. Here we see three different γ-ray energies, E1, E2 and E3 arranged so that we can obtain coincidence relations between them. Gating on energyE1 would mean to find the energies corresponding to E1 on one axis, and let the other axis show two different energies,E₂ andE₃for the two counts onE₁. From this we can know that E₁ is in coincidence with bothE₂ andE₃. It also goes the other way, as gating on E₂ or E₃ will give us E₁. Note however that E₂ and E₃ are not in coincidence with each other. This implies that the decay path cannot include both E₂ and E₃, so they must be parallel or belong to different paths of the level scheme. Based on this simpleγγ-matrix analysis, we can obtain the level scheme presented in Figure 3.9b.

3.3 Data analysis

3.3.1 Reproducing the half-lives of

Ni and

Ni

With the newly obtained data sets for theβ-decay of⁷⁶Ni and⁷⁴Ni, the first step was to reproduce their half-lives in order to know that all data sorting was done correctly and that the decay of the Ni nuclei was well understood. In reference [29], the half-lives of a wide range of exotic nuclei around⁷⁸Ni, including those of⁷⁴Cu and⁷⁶Cu, were obtained from the same experiment.

Several of the sorting and data-analysis codes used for the present data were adapted from codes by the author of Ref. [32], as very similar cases were studied. However, changes had to be made in order to work properly for the current data. In addition to the obvious changes like mass and making sure we are studying the correct nuclei,⁷⁶Cu and⁷⁷Cu have some important differences that needed to be included in the codes. The most important change was to add the isomeric state of⁷⁶Cu in the analysis.

The graph in Figure 3.10 shows the number of detected ⁷⁶Ni (violet) and

74Ni (green) ions in each DSSSD layer, while Figure 3.11 shows the velocity distribution of the detected ions in DSSSD layers 6 and 7. Knowing the velocity distribution for all DSSSDs of the desired nucleus could help to clean the data and remove contamination. This was done by performing a Gaussian fit for the distribution in each layer, and use this to determine boundaries that could be included in other codes.

Another observation from Figure 3.10 to note is the slightly counter-intuitive ratio of⁷⁴Ni compared to⁷⁶Ni. ⁷⁴Ni is less exotic and more likely to be produced in the fission reaction and we could expect there to be more⁷⁴Ni than⁷⁶Ni. The reason for why we see so much more⁷⁶Ni lies in the experimental setup which was tuned for very exotic nuclei near⁷⁸Ni (experiment 1) and⁸¹Cu (experiment 2). It is likely that many created ⁷⁴Ni ions never reached the decay station, as they were on the edge of the BigRIPS acceptance.

DSSSD number

0 1 2 3 4 5 6 7 8

Number of ions stopped in layer

0 50 100 150 200 250 300 350 400 450

103

×

Figure 3.10: Number of implanted ions of⁷⁶Ni (violet) and⁷⁴Ni (green) in the eight different DSSSDs. Two main points can be noted from this graph. One is that there is clearly more statistics for⁷⁶Ni, and the other is how the lighter ions of⁷⁴Ni go through more detector material before they stop.