CERN-THESIS-2017-151 29/08/2017 eeedr .Gins W. Neyens Begeleider: G. dr. Prof. Promotor: Spectroscopy Laser Collinear using chain isotopic aluminium along Moments Nuclear AUTI WETENSCHAPPEN FACULTEIT atro cec nd Fysica de in Science of Master reshitigdedtthet tot ingediend Proefschrift eae a egadvan graad de van behalen cdmear2016-2017 Academiejaar itrSMOLDERS Pieter Dankwoord

Aangezien deze thesis alleen maar tot stand kon komen dankzij het onderzoek dat in teamverband is uitgevoerd en de hulp die ik gekregen heb van de mensen die achter dit onderzoek stonden, wil ik eerst hier mijn dankbetuigingen geven. Eerst en vooral wil ik mijn promoter in Leuven, Prof. Dr. Gerda Neyens, bedanken om mij de kans te geven om in de zomer van 2016 het experiment bij te wonen in CERN, Geneve. Ik vond het bijzonder leerrijk en verrijkend om daar aanwezig te zijn en te observeren hoe een wetenschappelijk experiment op hoog niveau wordt uitgevoerd. Hierbij wil ik ook mijn begeleider Wouter Gins bedanken die mij wegwijs heeft gemaakt in Geneve en mij geholpen heeft met al de voorschriften om binnen te geraken. Daarnaast bedank ik hem ook voor de begeleiding in Leuven en voor alle technische hulp met het analyse programma SATLAS.

Many thanks also to all the phd-students of the COLLAPS collaboration that were present during the measurements. Thanks also to Livio who was also for the first time in CERN and with whom I spent some time going to all the mandatory safety courses. Special thanks go out to the people of the morning shift whom I had the chance to get to know better. This includes Ronald who had interesting views on electrical sockets from around the world. Also Xu that accompanied me during the morning commute from our residence and who also helped me in Leuven with the shell model calculations. And let’s not forget the rest of the morning crew: Laura, Simon and Stephan.

Last but not least I like to thank everyone from the nuclear moments group under Prof. Neyens in Leuven: Wouter, Xu, Ruben and Agi. They were always ready to help and give good advice during the progress meetings and helped me complete the analysis and this thesis.

i Abstract

The study of nuclei began with an experiment of Rutherford in 1911. Here, it was shown that the mass of an atom is located inside a small nucleus. This nucleus is made up of protons and neutrons that are arranged in a shell like pattern. This discovery led to the emergence of the nuclear shell model to identify and characterized nuclei.

Research into the nuclear structure and the shell configuration that the neutrons and protons adapt can provide information about one of the fundamental forces in nature, the strong force. This force binds nuclear particles into one structure, the nucleus, and determines the effective interactions between neutron and proton. Over the years, there have been developed multiple nuclear shell models to make predictions about the prop- erties of nuclei in different regions of the nuclear chart. These models are based on the interactions between the nucleons. A better understanding of the strong force can thus lead to better shell models.

It is predicted there are approximately 6000 different nuclei in the universe. Only a fraction of this amount has been investigated and most are located close to the valley of stability on the nuclear chart. This means the shell models that are being used at the moment are mainly based on the behaviour of the strong force in stable nuclei. During Research into the so called exotic nuclei, deviations between experiment and shell model predictions have been discovered. This means that the interactions between nucleons are different for these nuclei than for stable nuclei. To elucidate the behaviour of the strong force in all regions of the nuclear chart and to create better effective interactions to be used in nuclear shell models, the study of exotic nuclei is of great importance.

An example of new strong force behaviour in exotic nuclei are the islands of inversion. One of these islands is located on the nuclear chart from Z=10 to Z=12 and around N=20. However the exact borders of this island has not entirely been determined. The nuclei located inside the island exhibit a different shell configuration than expected by standard shell models for nuclei in that area of the nuclear chart. Normally, an energy gap is present between N=20 and N=21. This energy gap identifies the transition from one shell to another. The neutron or proton number at which a shell is filled is called a magic number. The standard shell model thus predicts a magic number at N=20. The nuclei inside the island of inversion on the other hand show a strongly reduced magicity of N=20 and therefore have a smaller energy gap. As a result these nuclei exhibit a ground state shell configuration that includes neutron excitations to the next shell across the gap. These are called intruder states.

This thesis studies the nuclear properties of a series of aluminium isotopes from 26Al to

ii iii

31Al by using Collinear Laser Spectroscopy (COLLAPS set-up). The magnetic dipole moment and electric quadrupole moment are determined. These moments are sensitive to the structure of the nucleus and can be used to check the validity of the normal shell model thereby seeing if there is any deviation in strong force behaviour. Specific attention will be given to 29Al and 30Al since both moments (magnetic and electric) are missing for 29Al and the electric quadrupole moment is missing for 30Al. Because aluminium has 13 protons and lies at the edge of the island of inversion, both these isotopes will be checked to exhibit the same deviation in strong force behaviour as isotopes belonging to island of inversion. Expectation is that they exhibit a normal ground state configuration since 31Al has a normal ground state, as confirmed in previous research.

From the results obtained in this research, 29Al and 30Al exhibit a normal 0p0h ground state configuration without intruder states. It can be concluded they are not part of the island of inversion. This conclusion was obtained by comparison of experimental nuclear moments with predictions made by the USD interaction (normal magicity) and the SDPF-M interaction (reduced magicity). Samenvatting

De studie naar kernen begon met het experiment van Rutherford in 1911. Hiermee werd aangetoond dat de massa van een atoom zich in een kleine kern bevindt. Deze kern bestaat uit protonen en neutronen die zich rangschikken in een schilpatroon. Deze ont- dekking leidde tot het ontstaan van het kernschilmodel om kernen te identificeren en karakteriseren.

Onderzoek naar de structuur van kernen en de schilconfiguratie waarin de neutronen en protonen zich bevinden kan informatie verschaffen over ´e´envan de fundamentele krachten, namelijk de sterke kracht. Deze kracht zorgt voor de binding van de kerndeeltjes tot ´e´en geheel, de kern, en bepaalt de effectieve interactie tussen neutron en proton. Over de jaren heen zijn er verschillende schilmodellen opgesteld die voorspellingen kunnen doen over de eigenschappen van kernen in verschillende regio’s op de nucleaire map. Deze modellen zijn gebaseerd op de effectieve interacties tussen de kerndeeltjes. Een beter inzicht in de sterke kracht kan dus leiden tot betere schilmodellen.

Er is voorspeld dat er 6000 verschillende kernen bestaan in het universum. Hiervan is nog maar een fractie daadwerkelijk onderzocht en deze kernen bevinden zich voornamelijk dicht bij de vallei van stabiliteit op de nucleaire map. Dit betekent dat de schilmodellen vooral gebaseerd zijn op het gedrag van de sterke kracht voor stabiele kernen. Bij on- derzoek naar exotische kernen zijn afwijkingen met de schilmodellen aangetoond, wat wil zeggen dat de interacties tussen de kerndeeltjes voor deze kernen anders zijn dan in de vallei van stabiliteit. Om het gedrag van de sterke kracht in alle gebieden van de nucleaire map te ontsluieren en om betere interacties voor het schilmodel te produceren, is de studie van deze exotische kernen van groot belang.

Een voorbeeld van dit afwijkend gedrag is het eiland van inversie. E´envan´ deze eilanden bevindt zich op de nucleaire map van Z=10 tot Z=12 en rond N=20, hoewel de exacte grenzen van dit gebied nog niet volledig bepaald zijn. De kernen in deze regio vertonen een heel andere schilconfiguratie dan voorspeld via het standaard schilmodel voor kernen in deze regio. Het normale schilmodel bevat een energiesprong tussen N=20 en N=21. Deze energiesprong identificeert de overgang van de ene schil naar de andere. Het neutron of proton getal waarbij dat de ene schil volledig gevuld is, noemt men een magic number. Het standaard schilmodel bevat dus een magic number bij N=20. Bij de kernen gelegen in het eiland van inversie is dit magic number sterk verzwakt en de energiesprong kleiner, zozeer dat de grondtoestand van deze kernen bestaat uit een configuratie van neutron excitatie tot de volgende schil.

Deze thesis zal aan de hand van de COLLAPS-setup (’Collineaire Laser Spectroscopie’)

iv v een reeks aluminium-isotopen bestuderen. Van 26Al tot31Al werd het magnetisch dipool moment (µ) en elektrisch quadrupool moment (Q) bepaald. Deze momenten zijn gevoelig aan de structuur van de kern en kunnen gebruikt worden om de validiteit van het schilmodel te controleren en te kijken naar mogelijke afwijkingen van de sterke kracht. Specifiek van 29Al zijn zowel het magnetisch dipool als electrisch quadrupool moment nog niet eerder bepaald, terwijl van 30Al het quadrupool moment niet gekend is. Aangezien aluminium met zijn 13 protonen aan de rand van het eiland van inversie ligt, wordt er nagegaan of deze 2 isotopen dezelfde afwijking vertonen. De verwachting is dat deze isotopen een normale grondtoestand vertonen aangezien voor 31Al al eerder bepaald werd dat het een gewone configuratie in de grondtoestand heeft [1].

Uit de vergelijking met de standaard USD interactie dat een normale energiesprong aan- neemt bij N=20 en de SDPF-M interactie die een verzwakte energiesprong aanneemt, blijkt dat 29Al en 30Al de standaard 0p0h grondtoestand vertonen met magic number N=20 en dus niet tot het eiland van inversie behoren. Contents

Dankwoord i

Abstract ii

Samenvatting iv

0 Introduction 1

1 Theory 2 1.1 Nuclear Shell model ...... 2 1.1.1 Shell structure evolution in exotic nuclei ...... 5 1.1.2 Island of inversion ...... 6 1.2 Nuclear electromagnetic moments ...... 8 1.2.1 Magnetic dipole moment ...... 8 1.2.2 Electric quadrupole moment ...... 9 1.3 Large Scale Shell Model calculations ...... 11 1.3.1 USD interaction ...... 11

2 Experiment 14 2.1 ISOLDE - isotope production ...... 14 2.2 Collinear Laser Spectroscopy ...... 17 2.2.1 Hyperfine structure ...... 17 2.2.2 COLLAPS ...... 19 2.3 Data ...... 24 2.3.1 Files ...... 24 2.3.2 P3/2 → S1/2 Transition ...... 25

3 Analysis 28 3.1 Voltage to Frequency conversion ...... 28 3.1.1 ISCOOL Voltage ...... 28 3.1.2 Fluke Voltage ...... 29 3.1.3 Scanning voltage ...... 30 3.1.4 Conversion ...... 31 3.2 SATLAS - data fitting ...... 31 3.2.1 BaseModel ...... 31 3.2.2 chi-squared fit routine ...... 32 3.2.3 Weighted average ...... 33 3.2.4 Basemodel optimization ...... 34

vi vii

3.2.5 Ratio Au /Al ...... 39

4 Results 40 4.1 Hyperfine Constants ...... 41 4.2 Nuclear Moments ...... 41 4.2.1 Magnetic Dipole Moment ...... 42 4.2.2 Electric Quadrupole Moment ...... 44 4.3 Comparison with Literature Methods ...... 46 4.4 New Results: 29Al and 30Al ...... 48

5 Conclusion 51

6 References 51 0 Introduction

The nuclei investigated in this thesis are aluminium isotopes from N=13 to N=18. Mea- surements on these isotopes were performed using the COLLAPS set-up. With this tech- nique, the hyperfine structure was determined from which the magnetic dipole and electric quadrupole moments were extracted. The isotopes for which there are no literature val- ues, namely 29Al and 30Al, are compared with theoretical predictions based on shell model calculations and can be subsequently identified as being part of the island of inversion or not.

The following is a description of the contents of the chapters.

Chapter 1 describes the nuclear shell model framework. This includes an introduction to the nuclear shell model and how it changes when handling exotic nuclei. An example of this change, the island of inversion, is also discussed. The nuclear properties, mag- netic dipole moment and electric quadrupole moment, are examined in detail. Finally a description of the large scale shell model calculations appropriate for the isotopes under investigation is given.

Chapter 2 describes the experiment. This includes a theoretical and technical review of the ISOLDE facility and COLLAPS-setup. It explains how the measurements are realized from creation of the isotopes of interest to the detection of the fluorescence photons leading to spectra in frequency space. The origin of the hyperfine structure visible on the spectra is also explained. Finally, the amount and contents of the raw data collected in the experiment are also discussed.

Chapter 3 discusses the analysis of the raw data and how the nuclear moments are obtained from the data files. First the conversion from voltage to frequency space summarized. Then, The SATLAS python package used for the fitting routine is discussed.

Chapter 4 summarizes the results obtained in this experiment and compares them with literature and shell model predictions. This chapter contains tables with hyperfine con- stants, magnetic dipole moments and electric quadrupole moments. The new nuclear moments for 29Al and 30Al are compared with USD and SDPF-M shell model calculations to investigate their ground state wave function.

Chapter 5 contains a conclusion to this research project. This includes a final assertion of the accuracy of the COLLAPS set-up for measuring nuclear moments and a confirmation of the position of 29Al and 30Al with respect to the island of inversion.

1 1 Theory

This chapter describes the manner in which the protons and neutrons are structured within a nucleus using the general shell model theory. The evolution of the shell structure for exotic nuclei is also discussed. Then an explanation of magnetic and quadrupole moment will be given together with their link to the hyperfine structure of a nucleus. Finally, the shell model calculations and specifically the USD interaction and its predictive power will be discussed in more detail.

1.1 Nuclear Shell model

The nuclear shell model came into existence by applying the principles of atomic theory to the nuclear realm. In atoms, the electrons moving around the nucleus occupy certain orbits at certain distances from the center of the atom. Each orbit can contain a certain amount of electrons based on the quantum numbers of that particular orbit and the Pauli exclusion principle, which states that no 2 fermions can have the same quantum numbers. The atomic properties are then determined by the electrons in the last unfilled shell, while the filled shells are considered an inert core. In terms of the nucleus, this translates to protons and neutrons moving in certain proton and neutron shells. Each shell contains a certain amount of nucleons depending on the quantum numbers of the nuclear states contained in that shell and also the Pauli exclusion principle that limits the amount to one nucleon for each different set of quantum numbers. The nuclear properties are determined primarily by the unpaired nucleon in the last unfilled shell. This analogy formed the foundation for the nuclear shell model.

Support for the validity of the nuclear shell model came through experimental observa- tions. One example is the change in nuclear charge radius with increasing amount of neutrons [2]. At certain neutron numbers a sudden increase in charge radius is observed as seen in Fig. 1.1, much like the increase in atomic radius when increasing the amount of electrons shown in Fig. 1.2.

2 3

Figure 1.1: Change in nuclear charge radius as a function of neutron number N. At the magic numbers for N, a discontinuity is seen. Picture taken from Ref. [2]

Figure 1.2: Change in atomic radius as a function of electron number. There are clear discontinuities in radius at transitions between electron orbits. Picture taken from Ref. [2]

This is evidence of the existence of nuclear shells that are filled at the neutron numbers at which there is a discontinuity of the charge radius. The nucleons inside these shells are expected to have a stronger binding energy which explains why they are spatially closer to each other compared to a nucleon in a subsequent shell, hence the increase in charge radius. The proton and neutron numbers at which a shell is filled are called magic numbers and are observed to be at Z, N= 2, 8, 20, 28, 50, 82, 126, ...

To create a nuclear model consistent with the observed shell structure, an appropriate nuclear potential had to be chosen. The main characteristics being the attractive central potential starting at intermediate distances and the repulsion at short distance. The Woods-Saxon potential (equation 1.1) fulfills these requirements [2]:

−V0 V (r) = (r−R) (1.1) 1 + exp[ a ]

This potential is then modified with a spin-orbit interaction of the form Vso = l · s, again based on the atomic model [2]. This additional term significantly alters the energy states of the nucleons. The interaction between spin s and angular momentum l splits the 4 energy levels and labels them with a new quantum number j = l + s and a degeneracy of 2j + 1. The effects of this additional interaction can be seen on Figure 1.3. On the figure, the spin-orbit correction has been applied to a harmonic oscillator potential instead of a Woods-Saxon potential.

Figure 1.3: Energy levels and their degeneracy obtained with a Harmonic Oscillator potential (left). Spin-orbit correction to energy levels (right). The shell structure is visible with the magic numbers in green. Picture taken from reference [3]

This adjusted potential can then be used in the Hamiltonian equation for the motion of a nucleon inside a nucleus governed by a mean potential [4]:

A X  p2  H = i + U (r ) (1.2) 2m i i i=1 i where A is the number of nucleons and Ui(ri) = V (r) + Vsol · s. Formula 1.2 is a simpli- fication of reality resulting from one of the fundamental assumptions of the shell model: every nucleon can be considered to be moving in a mean potential caused by all other nucleons. Solving the Hamiltonian equation gives the energy eigenstates with their respective de- generacy. Figure 1.3 shows the resulting levels and the shell structure with the correct magic numbers according to experiment are obtained. These levels represent the proton and neutron shells alike, however the proton shells will be shifted to higher energy due to the charge repulsion between protons. Each level is labeled with the quantum numbers n, l and j. Making the analogy with atomic theory again, the energy levels labeled by quantum number j can be considered as orbits on which the nucleons move about in the 5

nucleus. The energy of these single particle levels are also referred to as the Single Particle Energies or SPE’s.

1.1.1 Shell structure evolution in exotic nuclei The shell model in its extreme particle form dictates that the configuration of a nucleus corresponds to filling the single particle levels with nucleons starting at the lowest level until A nucleons are exhausted. The properties of the nucleus are then determined by the SPE in which the last unpaired nucleon is located [2]. For example, 41Ca has only one neutron beyond doubly closed shell (Z=20, N=20). The first level above this closed shell 7 is f7/2, so the nucleus will have a nuclear spin I equal to the total angular momentum j= 2 of the last unpaired nucleon. Doubly closed 40Ca will have a ground state spin of 0 since all nucleons are paired. To create an excited state in this nucleus, a pair of nucleons has to be broken to excite one of them to the next shell. So, a shell gap has to be crossed and the properties of the excited state is then determined by the coupling of the 2 odd nucleons. This method works well when evaluating nuclei close to doubly closed shell structures, but when nuclei with large unclosed shells are tested where there is space for the nucleons to interact, this method starts to break down. To see why, a closer look has to made at the nuclear Hamiltonian.

The Hamiltonian given by equation 1.2 doesn’t take into account the residual two-body nucleon-nucleon interactions. The full Hamiltonian is given by [5]:

X 1 X H =  a† a + < αβ|V |γδ > a† a† a a (1.3) a α α 4 α β δ γ In contrast to equation 1.2 which is written with the first quantization operator p, equa- tion 1.3 is written using the second quantization creation and annihilation operators, † respectively a and a. a are the SPE’s and the two-body nucleon interactions are ex- pressed by the Two-Body Matrix Elements or TBME < αβ|V |γδ >. The second part of equation 1.3 removes 2 particles from the states γ and δ and adds 2 particles in the states α and β. The matrix element governs this interaction. This additional part to the nuclear Hamiltonian causes changes in the SPE’s when moving away from stable isotopes and towards the drip lines on the nuclear chart. This is why it is important to consider the TBME’s when evaluating exotic nuclei. The two-body residual interaction potential V contains the different multipole interac- tions between nucleons. In the next section, the monopole part of this interaction will be discussed in more detail.

Monopole interaction The monopole part is used to create effective single particle energies (ESPE’s) [5]. The ESPE is defined as the sum of the SPE’s a and the monopole residual interaction. It has been proposed by T. Otsuka that the monopole interaction is produced by the tensor part of the nuclear force between protons and neutrons [6]. As an example of how this interaction can cause changes in the SPE’s, take a look at Figure 1.4 where the manner at how the neutron occupation of orbit j0 affects the energy of the proton orbit j is clarified. 6

Figure 1.4: Tensor interaction between neutron in orbit j0 and proton in orbit j. Picture taken from reference [6]

A neutron in orbit j0 interacts with a proton in orbit j via the monopole part of the residual nuclear interaction potential Vm. This causes an energy shift of orbit j equal to 1 ∆(j) = (V T =0(j, j0) + V T =1(j, j0))n(j0) (1.4) 2 m m where n(j0) is the occupation number of neutron orbit j0 and T refers to the different isospin states the proton-neutron system can have.[6]. Formula 1.4 shows that a larger occupation of j0 results in a stronger tensor interaction and thus energy shift of the orbit j. An explanation of determining in which direction the level is shifted lies in the relative 0 0 angular momentum of the two nucleons. If a neutron in orbit j< = l − 1/2 interacts with a proton in orbit j> = l + 1/2, then the large relative momentum causes the spatial wave function of the 2-nucleon system to be narrowly distributed. On the other hand, if orbit 0 j< interacts with orbit j<, then the relative momentum will be small due to same rotation direction and the wave function will be spatially stretched. The first case describes an 0 attractive interaction and thus when a nucleon in orbit j< interacts with nucleon in orbit 0 j> or j> interacts with j< the energy level of orbit j will be shifted down. The second case corresponds with a repulsive interaction and will result in an upward shifted energy of orbit j.

It is apparent that this effect can cause a gradual reduction of a shell gap. For example, if one increases the amount of neutrons which corresponds to increasing the exotic nature of a nucleus. This explains how the shell gap evolves for exotic nuclei.

1.1.2 Island of inversion There are multiple so called islands of inversion on the nuclear chart. These regions contain nuclei where the ground state configuration including excitations across a shell gap is energetically favorable compared with the normal ground state configuration. This is an inverted situation because normally excitations across a shell gap causes a drastic increase in energy, hence the name ’island of inversion’. Figure 1.5 shows one of these regions for nuclei with atomic number Z = 10, 11, 12 and neutron number around N = 20. 7

Figure 1.5: Nuclear Chart on which the island of inversion around N = 20 is indicated. Nuclei are labeled according to their ground state configuration. Picture taken from reference [7]

As seen in section 1.1.1, the location of the shell gaps can change for exotic nuclei, as is the case for these neutron rich nuclei. Instead of exhibiting the normal magic number N = 20, the neutron level vd3/2 is strongly shifted upwards towards the vf7/2 and vp3/2 levels. This causes the shell gap to disappear between these levels at N = 20 and creates a gap at N = 16 between the vs1/2 and vd3/2 levels. This process can be seen on Figure 24 30 24 1.6 for a O nucleus. In Si, 6 protons occupy the πd5/2 level. In O this proton level is unfilled and the attractive tensor force between protons in the πd5/2 level and neutrons in the vd3/2 level is gone which causes the shift upwards [8].

Figure 1.6: Left: The shell structure for stable 30Si with magic number N = 20. Right: shell structure of 24O shows emergence of new magic number N = 16 as a result of tensor interaction. Picture taken from reference [8]

For the nuclei that lie in the island of inversion this effect is less pronounced due to the presence of protons in the πd5/2 level, but it will still cause the magic number N = 20 to weaken and will facilitate excitations across this shell gap. These excitations cause the presence of deformed 2p2h intruder states in the ground state wave function of the nucleus that are lower or comparable in energy to the predicted spherical 0p0h state by the standard shell model theory [7]. For nuclei with closed shells where there are no available orbits for the nucleons to move 8 to, except for the ones across the shell gap, the ground state configuration is expected to be spherical. This is because an excitation across the shell gap would normally require so much energy that it would be near impossible to have a ground state configuration showing such an excitation. But for the nuclei in the island of inversion, this is feasible and the ground state exhibits a superposition of 2 configurations. One in which there is no nucleon excitation (0p0h) and one in which there are 2 neutrons that are excited from the sd-shell to the pf-shell [7]. This superposition of configurations is called configuration mixing. It originates from valence nucleons outside the closed shells that scatter each other into other orbits [3].

Looking back at Figure 1.5, the nuclei in and around the island of inversion are labeled according to the configuration of their ground state. The red label pf means that for these nuclei, the intruder state is actually lower in energy than the 0p0h state. It can be seen that the borders of the island of inversion are not well defined, so it is important to keep investigating the surrounding nuclei. In the following sections it is shown how we can probe the ground state configuration and identify intruder states by looking at the nuclear moments.

1.2 Nuclear electromagnetic moments

One way of looking at a nucleus is as a distribution of charges. This produces electric and magnetic fields inside the nucleus to which we can assign electromagnetic multipole moments. For example a spherical point charge, like a single electron, would exhibit only a monopole field. A more complex nucleus will have a more intricate structure, but will still tend to symmetric shapes. This is why it is sufficient to consider only the lowest order multipole moments when examining the electromagnetic properties of the nucleus [9]. The following sections will go into detail for two such multipole moments.

1.2.1 Magnetic dipole moment The classical definition of the magnetic dipole moment for a charge moving in orbit is [9] e |µ| = |l| (1.5) 2m where l is the angular momentum and e and m the charge and mass of the moving particle. When the quantum mechanical analog of this equation is taken, the magnetic moment is replaced by a magnetic moment operatorµ ˆ. The observable moment µ is defined along the direction of maximal angular momentum. Thus |l| has to be replaced by ml~ = l~, which is the expectation value of the angular momentum operator along maximum projection direction. The quantum mechanical counterpart of equation 1.5 is given by the following equation. [9] e µ = µ = ~ l (1.6) z 2m Besides an orbital angular momentum l, nucleons also posses an intrinsic spin s. This spin also creates a magnetic field and will also give a contribution to the magnetic moment. The following equations represent the two contributions [9]. 9

µ = gllµN µ = gssµN π π gl = 1 gs = 5.5856912(22) (1.7) v v gl = 0 gs = −3.8260837(18) gl and gs are called the g-factors. The values above are determined for free protons and e~ neutrons. µN is called the nuclear magneton and equals the quantity 2m when e and m are the proton charge and proton mass.

To evaluate the magnetic dipole moment in a nucleus, first note that nucleons inside closed shells do not contribute to µ. This is due to the pairing that results in a total angular momentum and spin of zero. Formula 1.8 considers all nucleons in orbits ji outside closed shells [10].

D n E X µ(I) = I(j1, j2, . . . , jn), m = I µˆz(i) I(j1, j2, . . . , jn), m = I (1.8) i=1

µˆz(i) is the magnetic dipole operator acting on each individual valence nucleon. The total magnetic moment of the nucleus is then the sum of these individual contributions sinceµ ˆ is a one-body operator. I is the nuclear spin and is determined by the orbits j occupied by the valence nucleons. It can be viewed as the analog of l in equation 1.5 for a free nucleon. m = I represents the maximal projection of nuclear spin just like ml = l in the free nucleon case.

When using equation 1.8 on a single nucleon outside closed shell, the Schmidt moments are obtained. These are single particle moments that can be calculated using the free nucleon g-factors of equation 1.7. [9]

 1 1 1 [(j − )gl + gs)]µN if j = l +  2 2 2 µ = (1.9) j(j+ 3 )  2 1 j 1 [ j+1 gl − 2 j+1 gs]µN if j = l − 2 The Schmidt moments of equation 1.9 can be used to calculate the total magnetic dipole moment for different possible configurations of a nucleus. However, it has been shown that the presence of other valence nucleons changes the single particle moments due to meson exchange interactions. The Schmidt moments have to be corrected for this effect before being used [10].

The magnetic moment of a nucleus is determined by the contributions of the valence nucleons, so its value is sensitive to the orbits that these nucleons occupy. The magnetic moment is thus a good probe for the purity of a configuration since the change of orbit occupation associated with a change in configuration will be reflected in the value of µ.

1.2.2 Electric quadrupole moment The electric quadrupole moment represents the deviation of the nuclear charge distribution from a spherical one. Its operator is defined as 10

Qˆ = e(3z2 − r2) (1.10) where e is the charge and z and r are the position coordinates of the orbit on which the nucleon moves [10]. Comparable to the magnetic dipole moment, the observable quadrupole moment is also defined as the evaluation of the operator along the direction in which the angular momentum has it maximal projection value. By convention the direction is chosen as the z-axis.

For a doubly closed nucleus where the protons and neutrons are all paired, we expect to find a spherical geometry. This means that the paired nucleons all move in spherically symmetric orbits for which the contributions to the quadrupole moment cancel each other. Thus the total quadrupole moment of a nucleus is determined by the contributions of the valence nucleons outside the doubly closed shells [9]. Exceptions do exist however: some heavy mass nuclei exhibit deformed equilibrium shapes due to collective effects of the nucleons inside the closed shells. These nuclei have an intrinsic quadrupole moment different from zero [2].

The contribution of a single nucleon located in orbit j is given by equation 1.11 of single particle quadrupole moments 2j − 1 Q = −e < r2 > (1.11) s.p. j 2(j + 1) j 2 where < rj > is the mean square radius of the orbit j [2]. When there are multiple valence nucleons occupying orbit j, the quadrupole contribution from that orbit is determined by  n − 1  Q (n) = Q 1 − 2 (1.12) j s.p. 2j − 1 where n is the occupation number of orbit j. Equation 1.12 shows that Q(particle) = - Q(hole).

Keeping in mind the single nucleon moments, now the total observable quadrupole mo- ment of a nucleus with spin I can be determined. This is done similarly as with the magnetic dipole moment: applying the moment operator Qˆ given by (1.10) to each in- dividual nucleon and summing the result. This is again possible since Qˆ is a one-body operator:

D n E X ˆ Qs(I) = I(j1, j2, . . . , jn), m = I Qz(i) I(j1, j2, . . . , jn), m = I (1.13) i=1 where n equals the number of valence nucleons [10]. Because equation 1.13 only considers valence nucleons, the free-nucleon charges eπ = 1 eff eff and ev = 0 have to be changed to effective charges eπ and ev . These effective charges will take into account the nucleon-nucleon and nucleon-core residual interactions that have been left out in equation 1.13. So neutrons, that are originally expected to have no contribution to the quadrupole moment because of their zero charge, now do contribute due to their non-zero effective charge. 11

The electric quadrupole moment is a good probe for the deformation of a nucleus. The intrinsic quadrupole moment associated with a deformed equilibrium state is given by:

3 2 Q0 = √ R Zβ(1 + 0.16β) (1.14) 5π av

where Z is the atomic number, Rav is the average nuclear charge radius and β is a deformation parameter related to the eccentricity of an ellipse. If β is zero, Q0 is zero as well and the nucleus has a spherical shape. [2] The intrinsic quadrupole moment can be determined by measuring the observable or spectroscopic quadrupole moment (Qs(I)). The link between these moments is given by: 3K2 − I(I + 1) Q = Q (1.15) s (I + 1)(2I + 3) 0 with K the projection of I along the axis of deformation of the nucleus. [10]

In conclusion, if a nucleus should exhibit a deformation, this will affect the value of the intrinsic quadrupole moment. The deformation can be caused for example by the presence of configuration mixing in which one of the configurations includes particle-hole excitations across a normally closed shell gap (as is the case with nuclei belonging to the island of inversion). By measuring the spectroscopic quadrupole moment, the amount of deformation and subsequently the amount of intruder state mixing in the ground state of nuclei can be determined.

1.3 Large Scale Shell Model calculations

In the independent particle approach to the shell model, it was assumed that the properties of a nucleus could be predicted by looking at the orbit occupied by the singly unpaired nucleon. In the previous sections, we have seen that this is simply not realistic due to the effects of residual nucleon-nucleon interactions. A new method had to be found that implements these interactions and where nucleons can be present in multiple orbits. The answer came in the form of shell model calculations that assume certain interactions in the form of TBME’s and implement them in certain model spaces in which the the possible configurations are calculated. The model spaces determine for which nuclei the properties can be predicted, but the steps taken in the calculations are the same regardless of the model space. For this reason the USD interaction will be used to illustrate the inner workings of large scale shell model calculations.

1.3.1 USD interaction The first step in interpreting shell model calculations is defining the model space. In case of the USD model, the shells up to the magic number Z,N = 8 are filled and are considered an inert core. This means that the nucleons in these orbits are assumed to be fixed and cannot excite to other levels. The valence space is the sd-shell above this inert core consisting of the 3 orbits d3/2, d5/2 and s1/2 that can be filled with valence nucleons [11]. Figure 1.7 visualizes the model space of the USD interaction. 12

Figure 1.7: Shell model space of the USD interaction. The valence configuration space is bounded by two shell gaps.

The nuclei that can be described by the USD model correspond to nuclei with atomic number Z and/or neutron number N between 8 and 20. It is assumed that for these nuclei, the valence particles will only occupy orbits in the sd-shell and will not be able to occupy orbits in the next pf-shell separated by a large shell gap. We’ve already seen that there are nuclei in the island of inversion that have configurations in which this is indeed the case and for these nuclei, a larger model space, including the pf-shell, has to be chosen. This expanded model space is described by the SDPF shell model calculations, but this will not be discussed in detail. For more information on the SDPF-interaction, see the work of Retamosa, Caurier, et al. in reference [12].

The next step is to solve the eigenvalue problem Hψ = Eψ with the nuclear Hamiltonian from equation 1.3, ψ being the wave function of the nucleus and E the energy levels. To do this, the matrix elements < φi|H|φj > have to be determined for all possible configurations φ of the valence nucleons. The configurations are represented by Slater determinants that are products of single particle states occupying different orbits. This matrix is then diagonalized to solve the eigenvalue problem [3]. Because the nuclear Hamiltonian is rotationally invariant, the matrix elements conserve quantum number J and M, corresponding to total angular momentum and its z-component. The matrix elements are thus determined by the possible orbit occupations of the valence nucleons or SPE’s and the TBME’s between the nucleons that also preserve total J and M. The energy eigenvalues equal:

E(J) = 1 + 2 + VJ (1.16) where i are the SPE’s of the orbits the two nucleons occupy and VJ correspond to the TBME’s of nucleons that couple to total angular momentum J and take on the form:

VJ = j1, j2, J, M V j3, j4, J, M (1.17) 13

where ji represents the nuclear orbit occupied by one of the nucleons and V is the residual nucleon-nucleon interaction part of the nuclear Hamiltonian. The valence nucleons can occupy any of the 3 orbits of the sd-shell. There are thus many matrix elements with different J-values to consider and for each J-value multiple TBME’s. This causes the matrix to be diagonalized to grow considerably with each added orbit to the valence space. In the case of the USD interaction, there are 3 SPE’s and 63 TBME’s [3].

The final step is comparing the calculated energy levels E of equation 1.16 with the experimentally determined ones. There will most likely be a discrepancy due to the unknown exact form of the residual interaction V . The interaction is calculated from basic meson exchange theory using the G-matrix formalism of Kuo [13]. Due to the discrepancy, a χ2 analysis is performed and the TBME’s are adjusted. This process is repeated until χ2 is minimized or reaches a certain minimum threshold [3].

Using the USD shell model calculations, the configurations that are part of the nuclear wave function can be determined and the orbits occupied by the valence nucleons are known. Using this information, many properties of the nucleus can be determined, e.g. magnetic dipole and electric quadrupole moment. By comparing experimental values of a nucleus to the USD predictions, it is possible to determine if the nucleus exhibits the normal magic numbers at Z,N=20 or if it breaks with the predictions of the USD model and belongs to the island of inversion.

Large scale shell model calculations are an essential tool for exploring the exotic regions of the nuclear chart due to its predictive power. However, in this exotic frontier, magic numbers can differ and shell gaps can disappear. For this reason, larger model spaces, often consisting of two shells, have to be considered to perform the calculations. This in- creases the dimensions of the Hamiltonian matrix so that the diagonalization is becoming computationally challenging for computer programs. To overcome this problem, a new method called Monte Carlo Shell Model (MCSM) has been developed [14]. This method reduces the dimension of the H-matrix by selecting only a few important Slater determinants (or configurations) as basis states. This reduced ma- trix is then diagonalized and has been shown to give similar results as the diagonalization of the entire matrix. With MCSM, the former computational limits can be crossed and dimensions as large as 1015 can be reached, as is shown in Figure 1.8. Figure 1.8: Upper limit on dimensions of Hamiltonian matrices that could be analyzed over the years. Blue dots refer to conventional diagonalization. Red dots refer to dimen- sions analyzed with Monte Carlo Shell Model. Picture taken from [3]

2 Experiment

The measurements of the aluminium isotopes were performed at the European Organiza- tion for Nuclear Research or CERN located in Geneva, Switzerland. More specifically they were performed at the ISOLDE facility during the summer months of the year 2016. The following sections will discuss the isotope production and subsequent measurement with the COLLAPS set-up. Also the analysis of the raw data and extraction of the hyperfine parameters from spectra will be described.

2.1 ISOLDE - isotope production

ISOLDE stands for Isotope Separator On-Line DEvice. In this building radioactive ion beams are produced that are used in a variety of experimental set-ups scattered around the beam lines. Figure 2.9 shows a schematic of the ISOLDE facility. The facility uses the high energy proton beams that originate from the Proton Synchrotron Booster (PSB) located nearby that produces protons with energies of 1.4 GeV. These protons bombard a target material and create an array of radioactive nuclei through nuclear reactions like fission, spallation and fragmentation [15]. To have a large yield of created nuclei, the target has to be chosen well as to ensure a fast release of nuclei. This is why the target material is heated between 700◦C and 1400◦C. Typical release times achieved are around 30 seconds. To achieve smaller release times down to tenths of a second, the target material must be heated to above 2000◦C[16]. The atomic identity of the target also determines which nuclei are created in this process. For the aluminium isotopes discussed in this paper, an uranium-carbide (UCx) target was used.

14 15

Figure 2.9: Schematic of the ISOLDE facility. The green line is the path the isotopes from this experiment traveled from creation to arrival at the COLLAPS set-up. Picture taken and adjusted from reference [25]

The released nuclei then diffuse to the ion source. ISOLDE uses three different types of sources: surface ion sources, plasma ion sources and laser ion sources. In this experiment, the laser ion source RILIS was used. RILIS is in charge of selecting the correct isotopes for the experiment amongst the nuclei that were released from the target. The name RILIS stands for Resonance Ionization Laser Ion Source and refers to the method by which it selects the isotopes. The ion source is nothing more than a hot cavity through which multiple lasers are sent to ionize a certain element through a step-wise resonant excita- tion. The lasers are tuned to frequencies that correspond with the successive electronic excitations of a specific element until this element is ionized. This method uses the unique atomic energy structure and has a large Z-selectivity [17]. The ionized nuclei are then accelerated away from the source by applying an electrostatic potential of 30 to 60 kV.

The selected isotopes then arrive at the analyzing magnet. ISOLDE currently has two magnet systems in use. The GPS uses one bending magnet at an angle of 60◦. The HRS or High Resolution Separator uses two dipole magnets, first one at 90◦ and the second at 60◦ angle. The analyzing magnet collects isotopes of certain mass at the focal plane of the magnet, while the isotopes of other masses are lost. The underlying principle is the change in orbiting radius of an ion feeling a magnetic field depending on the mass of the ion. During the measurements, different masses could be selected by increasing or decreasing the magnetic field of the analyzing magnet. In this research project, the HRS is used with its superior mass resolving power M/∆M between 7000 and 15000 compared to the GPS system with resolving power of only 2400. [18]

The isotopes collected at the focal plane of the magnet system are subsequently cooled and possibly bunched by a gas-filled linear Paul trap called ISCOOL. Figure 2.10 shows a diagram of the RF-potential used in the trap. ISCOOl consists of four quadrupole rods 16

Table 2.1: List of Al-isotopes investigated in this paper together with their half-life and isotope yield. The yields are all determined for UCx target and RILIS ion source. Data taken from ISOLDE yield database [20].

isotope half life yield (ions/C) 26Al 7.2E5 y not listed 26mAl 6.3 s 6.8E4 27Al stable not listed 28Al 2.2 min 4.0E7 29Al 6.6 min 4.5E7 30Al 3.6 s 2.5E6 31Al 644 ms 2.5E5 pairwise coupled to an applied radio-frequency voltage that confine the ions in the radial direction. Surrounding these rods are segmented DC electrodes that create a potential gradient guiding the ions to the end of the ISCOOL ’tube’. The structure is also filled with Helium gas at a pressure of 0.1 mbar that slows down the ions and reduces their energy spread to about 1 eV through thermal collisions [19].

Figure 2.10: Schematic of the ISCOOL Paul trap. Picture taken from reference [19]

ISCOOL can also be used to accumulate the ions together by applying a temporal trapping potential. In this way, bunches of ions can be released in succession instead of a continuous stream of ion release. By changing the accumulation time, the amount of ions in one bunch and the time window between releases can be determined by preference. In this experiment, the bunched beam mode was used for reasons that are linked to the operation of the COLLAPS set-up and will be discussed later on.

The isotope yields that are achieved in the ISOLDE facility depend on the target used and the ion source, assuming that the energy of the incoming protons is fixed. There will be occasional losses of ions along the transfer lines through which the ions are transported, but these are minimized by using quadrupoles for focusing of the ion beam. In this experiment the target is UCx and the ion source is RILIS. The yield of the aluminium isotopes achieved hereby is listed in table 2.1. 17

2.2 Collinear Laser Spectroscopy

Laser spectroscopy is a technique that probes the atomic structure and provides infor- mation on the electronic states of an atom. A laser beam is used to excite atoms by matching the frequency of the laser with the energy of one of the atomic transitions. From the examination of the atomic properties, nuclear properties can be extracted. This is because the hyperfine interaction between the atomic electrons and the nucleus changes the atomic structure. This effect is studied. To detect this change, a high resolution is required. Collinear Laser Spectroscopy or COLLAPS fulfills this requirement. In this section we shall see how this technique provides the desired results.

2.2.1 Hyperfine structure The atomic structure of an atom is determined by the electron occupation of the orbits surrounding the nucleus. An atomic transition happens when an electron moves from one orbit to another. These configurations of orbit occupation determine the electronic states. Suppose now that the nucleus is a point charge. The atomic structure would be defined by an electronic Hamiltonian:

ˆ ˆ ˆ He = Te + Uee (2.18) − − where Te represents the kinetic energy and Uee the e - e Coulomb interaction, together with a monopole interaction between the total charge of the nucleus and the electrons given by:

2 ˆ0 ˆ 0 −e NZ Q · V = Une = (2.19) 4π0re

Here −eN is the charge of the electron cloud, eZ the charge of the nucleus and re the distance between nucleus and electrons. Qˆ0 ·Vˆ 0 represents the monopole part of the inter- action between the electrostatic multipole moment Qˆ of the nucleus and the electrostatic multipole field of the electron cloud generated at the nucleus Vˆ [21]. The electronic levels obtained by these interactions is called the fine structure of the atom.

In section 1.2 we saw that the nucleus also has higher multipole moments reflecting its internal structure. There are two different interactions as a consequence of this structure. The first is the non-spherical charge distribution of the nucleus which alters the Coulomb electrostatic potential between nucleus and electrons (charge-charge interaction). The other is the movement and the spin of the nucleons that create respectively convection and spin currents inside the nucleus. These nuclear currents can interact with currents generated by the electron cloud (current-current interaction).

The Hamiltonian for the current-current interaction between a single electron and a nu- cleus is [21]:

∞ X B(n) · M (n) Hˆ = (2.20) jj 2n + 1 n=0 18

(n) j in Hjj refers to the current density distribution, M is the magnetic multipole moment of the nucleus of order n and B(n) is the magnetic multipole field at the nucleus generated by the electron cloud. Due to the parity requirement of the nuclear state |I >, only odd n terms will contribute to equation 2.20. This means that only the dipole, octupole, etc. contributions remain. Assuming only the dipole term has a significant contribution (higher order terms are increasingly less significant), M (1) can be defined as the magnetic (1) dipole moment of the nucleus µI , B defined as the magnetic hyperfine field B(0) and the current-current interaction rewritten as:

ˆ ˆ Hjj = −µˆI · B(0) (2.21) Considering an atomic system including hyperfine interactions, the relative orientation of nuclear angular momentum I and electronic angular momentum J has an effect on the energy state of the atom. The atom can be described by a new quantum number F which is the vector sum of I and J and takes on values from a parallel F = I +J to anti-parallel F = |I − J| orientation. Computing the matrix elements of the Hamiltonian of equation 2.21 on an atom in a state |F > results in the energy contribution of the current-current interaction: 1 < F |Hˆ |F >= E = A C (2.22) jj jj 2 J where C is a shorthand notation for F (F +1)−I(I +1)−J(J +1) and AJ is the magnetic hyperfine coupling constant:

µB(0) A = (2.23) J IJ

The charge-charge interaction between the nuclear and electronic charge distribution also has an effect on the energy state of the atom. Equation 2.19 already gave the monopole contribution. In general, the Hamiltonian for the charge-charge interaction is given by:

∞ X Qˆ(n) · Vˆ (n) Hˆ = (2.24) qq 2n + 1 n=0 Due to the parity requirement of the nucleus, only even n terms survive. This corresponds to contributions of the monopole term (n = 0) and quadrupole term (n = 2), assuming higher order terms are negligible. Evaluating the Hamiltonian of equation 2.24 for nuclear states |F > gives the charge-charge interaction energy contribution:

3C(C + 1) − 4I(I + 1)J(J + 1) < F |Hˆ |F >= E = B (2.25) qq qq J 8I(2I − 1)J(2J − 1) where BJ is the electric hyperfine constant:

BJ = eQVzz (2.26)

Q is the electric quadrupole moment of the nucleus and Vzz is the electric field gradient at the nucleus generated by the electron cloud. From the definition of the quadrupole moment given in section 1.2.2, it is known that Q is zero for nuclei with a spherical charge distribution (when I = 0 or I = 1/2). Also when the electronic spin J equals 0 or 1/2, 19

the electron cloud has a spherical symmetry. This means that the electric field gradient Vzz will vanish. Thus, the charge-charge interaction energy given in equation 2.25 will only contribute if I, J > 1/2 [21].

Figure 2.11: Schematic of atomic energy levels and influence of fine and hyperfine inter- action. The hyperfine structure is shown for a nuclear spin I = 1/2. Energy scales are added in green.

Both the magnetic dipole and electric quadrupole interaction depend on the relative ori- entation of nuclear spin I and electronic spin J. It is this interaction that changes the fine structure of the atom, characterized by quantum number J, into the hyperfine struc- ture characterized by quantum number F . Figure 2.11 shows schematically the difference between fine and hyperfine structure with dummy values for I and J. The fine structure levels undergo splitting as a result of the different orientations of I and J and the amount of splitting is determined by equations 2.22 and 2.25 together. The energy scale of the hyperfine splitting lies in the µeV range, while the fine structure splitting lies in the meV range [21].

2.2.2 COLLAPS After the bunching and cooling by ISCOOL, the ion beam is accelerated to an energy of tens of keV before being sent to the COLLAPS set-up. The availability of such an ion beam provided by the ISOLDE facility is well suited for high-resolution laser spectroscopy such as the collinear laser spectroscopy method. In this method, the ion beam is collinearly overlapped with a co-propagating laser beam. Photons from the laser are absorbed by the ions if their energy matches an atomic transition. This resonance can be detected by the fluorescence photons emitted by the ions after decaying back to a lower-lying level. By sweeping the laser frequency across a certain atomic transition, the hyperfine structure of the ions can be probed. This process is schematically shown on Figure 2.12. 20

Figure 2.12: Schematic of the basic operating principle of the COLLAPS set-up.

Spectral line width The frequency difference between transitions of a hyperfine structure is typically between 10 and 1000 MHz. To be able to distinguish between the obtained spectral lines of such a structure, a very high resolution is needed [22]. The key to obtain this resolution, is the acceleration of the ion beam. The line width of the spectral lines can be calculated from the velocity distribution of the ions. Before arriving at COLLAPS, the ions have an energy spread δE of around 1 eV, thanks to ISCOOL. The velocity distribution of the ions can be determined from the kinetic energy spread by the following equation [23]. 1 δE = δ mv2
= mvδv (2.27) 2 An acceleration voltage V gives an energy of eV to the ions. This acceleration will be given to all ions equally, so the energy spread δE remains the same. Equation 2.27 shows that the velocity distribution δv decreases if the velocity increases. The acceleration is thus the determining factor in decreasing the velocity distribution and also the line width as will be shown next. The relative motion between co-propagating beams of photons and ions causes a Doppler- shift of the photon frequency in the rest frame of the ions. An approximation of the Doppler-shift is given by ∆f = f0(v/c) where f0 equals the laser frequency. Due to the velocity distribution, the ions will absorb the photons at different frequencies due to the different Doppler-shifts for each velocity v. This frequency range determines the line width of the detected fluorescence photons and is called Doppler broadening. The line width or Doppler width is given by δf = f0(δv/c). Combining this equation with equation 2.27 and using the relation v = p2eV/m, the following expression for Doppler width is obtained: δE δf = f0 √ (2.28) 2eV mc2 where m is the mass of the ions. Realistic values inserted in formula 2.28 results in a Doppler width of around 10 MHz, which is small enough to probe the hyperfine structure [23]. An exact analysis of the line width obtained in this experiment will be given in chapter 3.

Besides the Doppler width, there are also other contributions to the line width. One 21 comes from the natural line width of the atomic transition that is excited by the laser. Due to the limited lifetime of an excited atomic level, the measured energy and frequency of the fluorescence photon will have an inherent uncertainty determined by Heisenberg uncertainty principle (δEδt > ~): δE 1 δf = = (2.29) nat h 2πτ with τ the lifetime of the excited state. Another contribution to the line width comes from power broadening. Opposite to the natural line width that is determined by spontaneous emission, power broadening comes from stimulated emission. When the laser power is increased, more ions will be excited and more fluorescence photons will be emitted. This increase in stimulated emission rate lowers the lifetime of the excited level compared to the natural situation with only spontaneous emission. This in turn causes an increase in line width according to Heisenberg’s principle. The following equation gives the line width due to power broadening in terms of the laser power P and saturation power Psat [24]:

r P δfpow = δf 1 + (2.30) Psat where δf is the line width in absence of power broadening.

Doppler-tuning To probe the hyperfine structure, the most straightforward way would be scanning the laser frequency across the atomic transition frequency. A more accurate method is to fix and stabilize the laser to the optical transition frequency and use the Doppler-shift to change the frequency experienced by the ion beam by varying the velocity of the ions. The exact equation for the Doppler-shifted laser frequency experienced by the ions is: 1 ∓ β f = f0 (2.31) p1 − β2 with β being the relativistic Lorentz factor. The minus sign corresponds with co-propagating beams while the plus sign corresponds with contra-propagating beams. β equals: s v m2c4 β = = 1 − (2.32) c (eV + mc2)2 This equation is obtained by using the relativistic energy of the ions: E2+(mvc)2 = (mc2)2 with E = eV + mc2. By applying a changeable acceleration voltage, β can be varied and thus the scanning frequency [23]. This Doppler-tuning is the standard operating mode of the COLLAPS set-up. 22

Set-up

Figure 2.13: Schematic of the components of the COLLAPS experimental set-up including the voltage supplies and electronic readouts. Picture taken from reference [25]

Figure 2.13 shows a schematic of the COLLAPS set-up. The overlap between ion and laser beam is managed by 10◦ electrostatic deflection plates. In the post-acceleration region, the ions receive a voltage from two sources. One is a signal from the DAC computer that can supply a voltage between −10 and +10 V used to scan the hyperfine structure. This voltage is amplified by a linear-voltage amplifier called Kepco. Another voltage is supplied by a high-voltage power supply called a Fluke. The Fluke voltage ranges between −10 and +10 kV and is used to match the atomic transition frequency with the laser frequency for the different masses of the measured isotopes [25].

After acceleration the ions are brought into the Charge Exchange Cell (CEC). The CEC is a heated, vapour filled chamber where the singly charged aluminium ions are neutralized trough collisions with the gas atoms. In these collisions, the atomic electron clouds of the 2 particles come into contact and one of the electrons from the gas atom is transferred to the ion, effectively neutralizing it. This neutralization is performed in cases where the ionic transitions are too high in energy to be reached with a continuous wave laser system. Equation 2.33 gives the reaction between aluminium ions and the sodium gas used in this experiment.

Al+ + Na → Al + Na+ + ∆E (2.33) The energy defect ∆ E is typically small (< 1 eV) so that the reaction occurs resonantly in energy and is characterized by a large cross section and no appreciable loss in momentum of the original beam [23]. 23

Besides the elastic collisions, there will also be a certain amount of inelastic collisions where the aluminium particles lose some of their energy. This side-effect causes the emergence of sidepeaks in the spectrum that are shifted with respect to the main resonance peaks. This will be discussed in more detail in chapter 3.

The final step is the detection of the fluorescence photons which is done by a system of 4 photomultiplier tubes or PMT’s. A photon that impacts the photocathode plate of the PMT, releases an electron through the photoelectric effect. This electron is multiplied inside the tube and the amplified signal is measured and stored. The main disadvantage of the PMT system used in the COLLAPS set-up is the low efficiency and especially the high number of background counts. This becomes troublesome when measuring exotic isotopes that have a low production yield. There are several sources for background counts, the following list will discuss them and solutions to suppress them if possible: • Unwanted photons can come from the interaction of the ions with stray gas atoms of residual air left in the COLLAPS apparatus. Since the entire set-up is kept at high vacuum (10−7 mbar), these photons have an insignificant contribution. Photons created by interaction with the gas atoms from the CEC will have a much larger effect on the background. Light of this kind, produced in an area other than the light collection region, can be suppressed by an aperture system. The laser and ion beam travel superimposed through the apparatus with a certain circumference. By placing an aperture of certain size between the CEC and light collection region that allows only the ion beam to pass, much of the stray light can be stopped before reaching the PMT detection system. The PMT system including the aperture is shown on Figure 2.14. The picture also depicts a cooled housing, which was removed in this experiment [25].

Figure 2.14: Schematic of the fluorescence detection system showing the 4 PMT’s, the aperture set and lens system. The cooled housing shown in this picture was removed in this experiment. Picture taken from reference [25]

• Another source of background photons is laser light scattered from components of 24

the apparatus and radioactivity collected inside the apparatus [23]. This kind of light can be produced inside the light collection region and thus cannot be stopped by the aperture. A way to reduce background counts in general is by operating ISCOOL in bunched mode. The PMT signal can be gated as to only allow photon counting when a bunch has reached the light collection region. In this way the temporal constant background can be greatly reduced by measuring in certain time intervals when fluorescence photons are expected. The accumulation time in ISCOOL is set to create bunched ions with a temporal width of ±20 µs and temporal spacing between bunches of ±100 ms. Background photons can thus only contribute for a period of a couple of microseconds. The background can be suppressed by a factor of 103 − 104 (100ms/20µs) [26].

• The PMT is kept in the dark at all times with a black blanket to ward off photons from the surroundings. Even if no voltage is applied and no measurements are being performed, photons that hit the photocathode will produce electrons. These electrons can influence measurements for a few days after.

• At last, there are the dark counts that increase the background. Dark counts are counts that the PMT detects when there are no photons hitting the phototcathode plate. They are mostly of thermal origin and are reduced by cooling the PMT’s.

Another way to improve the set-up is not to suppress background, but to improve ef- ficiency. This is done by using a lens and spacer system in the detection geometry of the PMT system (see Figure 2.14). The lenses focus the emitted fluorescence photons to impinge on the photocathode plate. This increases the probability of a photon reaching the plate. The location of the focal point between the lens and the PMT depends on the wavelength of the fluorescence photon. For practical reasons, the same lens should be used for every experiment [25].

2.3 Data

2.3.1 Files The spectra of fluorescence counts collected by the COLLAPS experiment are gathered and saved by the Measurement Control Program or MCP. The MCP-files together with the lab book contain the raw data and all the information of the experiment necessary to perform an analysis. Table 2.2 shows an example for the measurement ’run234’ of 27Al of the information that can be extracted from the MCP-files. The photocounts are saved for each PMT, gated and ungated. The photocounts used in the analysis is the sum of the gated photocounts collected by the 4 PMT’s. The spectra for which there were unstable parameters, e.g. drift in laser frequency or fluke voltage, are left out of the analysis. Sometimes a sequence of measurements are performed on top of the previous MCP file to increase statistics. This is indicated by ’Go’. Of such a sequence, only the last MCP file is used for analysis. Table 2.3 contains the total amount of MCP files for each isotope and the amount of files that were used for the final analysis discussed in chapter 3. 25

Table 2.2: Parameters extracted from MCP file useful for the analysis. Example values are shown for run234 of 27Al.

Parameters from MCP file values for run 234 of 27Al Line Voltage (V) (min. − max.) -2.5 − 0.5 Photocounts (#) (min. − max.) 34 − 1414 Fluke (#) 1 Fluke Voltage (V) 668.99698 ISCOOL Voltage (V) 29976.41806 Timestamp 07/08/2016 20 : 27 : 57

Table 2.3: Amount of MCP files gathered and files used for analysis for each isotope.

Isotope # MCP files (total) # MCP files (analysis) 26Al 43 17 27Al 97 70 28Al 8 7 29Al 10 10 30Al 10 4 31Al 15 3

2.3.2P 3/2 → S1/2 Transition The atomic transition of aluminium measured in this experiment is the transition from the 3P3/2 to the 4S1/2 level. Figure 2.15 shows a simplified level scheme of aluminium and the wavelengths of the different possible transitions.

Figure 2.15: Level scheme of aluminium atom. Picture taken from reference [27].

With its 13 electrons, aluminium occupies the electronic 3P level in its ground state. The P3/2 level is chosen because it is sensitive to the quadrupole moments of the Al-isotopes (J > 1/2), while the P1/2 level is not. The P3/2 - S1/2 transition also has a higher Einstein 8 7 co¨efficient (A = 1 · 10 ) than the P1/2 - S1/2 transition (A = 5 · 10 ) and is therefore a 26 stronger transition. The transition to the 2D doublet is also not preferred due to the proximity of the two levels. This would cause mixing of the two transitions [27]. The transition of interest has a wavelength of 396 nm. A Matisse Ti:Sa laser with light of 792 nm is frequency doubled by the process of second harmonic generation to produce light of 396 nm used to excite the aluminium isotopes.

Figure 2.16 shows the transition between the two atomic levels including the hyperfine splitting that is probed by the COLLAPS method. The amount of splitting is also in- dicated and calculated using equations 2.25 and 2.22. The calculation was done for the reference isotope 27Al with nuclear spin I = 5/2. The transitions between the hyperfine levels are matched with the observed fluorescence peaks in a collected spectrum. The allowed transitions between the hyperfine F levels are determined by the selection rules of quantum number J, namely the electric dipole radiation (E1) of electron excitation:

∆F = 0, ±1 F = 0 6→ 0 (2.34) 27

Figure 2.16: Hyperfine structure of the J = 3/2 to J = 1/2 atomic transition in 27Al (I = 5/2). The transitions are matched with the peaks in the experimental spectrum shown underneath. 3 Analysis

The voltage in the MCP files is first converted to the observed frequency. Afterward the analysis is performed using a custom written Python package called SATLAS (’Statistical Analysis Toolbox for LAser Spectroscopy’). It was created by PhD students Wouter Gins and Ruben de Groote at the Institute for Nuclear and Radiation physics in Leuven [28]. The program provides an interface for normal fitting routines and adapted equations for the analysis of counting data. 3.1 Voltage to Frequency conversion

The ions receive multiple voltage accelerations along their path from creation to detection. Equation 3.35 shows the total voltage experienced by the ions:

Vtot = VISCOOL − Vfluke − k ∗ Vscan (3.35) where k is the kepco amplification factor. The voltage that the ions receive at ISCOOL can be viewed as a downward slope along which the ions slide off and gain energy as can be seen on Fig. 2.10. In contrast to this, the fluke and scanning voltage can be viewed as barriers. This explains the opposite signs in equation 3.35.

3.1.1 ISCOOL Voltage The voltage supplied to ISCOOL can be obtained from the MCP files for each measure- ment. Figure 3.17 shows the stability of the voltage during the experiment.

Figure 3.17: Stability of ISCOOL voltage during the experiment.

28 29

At one point, the pressure in ISCOOL was changed. This caused the voltage to drop. The measurements continued before the voltage had settled to its original value. Excluding these values, the voltage is stable throughout the experiment, lasting several days, with a maximal variation of 1.4 volt.

3.1.2 Fluke Voltage Because the laser frequency is fixed, the ions have to be accelerated to match their atomic transition frequency with the laser frequency (see section 2.2.2). This is done by a voltage supplied by the fluke. There were 3 flukes available for this experiment, but since fluke 2 showed discrepancy between the read-out and the actual supplied voltage, only fluke 1 and 3 were used for measurements. Figure 3.18 shows the stability of fluke 1 for the different masses of the aluminium isotopes. Some figures show plateaus of stable voltages. These signify subsequent measurements that are interrupted by measurements of another mass. When the original mass is measured again, the fluke voltage setting can be shifted by a small amount. The large variation (3 V) for 26Al is due to a change in voltage setting on the fluke. The change in voltage is a result of the change in laser frequency lock that took place between the measurement sequences of 26Al. It can be concluded that the fluke is stable during subsequent measurements and shows a variation of the order of 0.01 V when this effect is neglected.

Figure 3.19 shows the stability of fluke 3. Also here plateaus of subsequent measurements are visible. Fluke 3 is also stable with maximal variation of 0.3 V.

Figure 3.18: Stability of fluke 1 voltage during the experiment. 30

Figure 3.19: Stability of fluke 3 voltage during the experiment.

3.1.3 Scanning voltage The scanning voltage supplied by the DAC is used to scan across the hyperfine structure. When a new isotope is measured, the scanning voltage is varied until the entire spectrum lies within the scanning voltage. This voltage is user dependent and can be found in the MCP-files. The scanning voltage is amplified by a kepcofactor k. Because the fluke is not grounded, it supplies a voltage to the beam line with respect to the amplified scanning voltage. Fluke devices are never 100% identical, so the amplification factor k is determined for each fluke separately. The kepcofactor k is determined by means of calibration curves or ’kepcoscans’ that measure the amplified voltage k·Vscan versus Vscan. Figure 3.20 shows such a calibration curve.

Figure 3.20: Calibration curve of fluke 1.

There were 9 kepcoscans taken during the experiment. The average of the determined slopes of all scans is taken as the kepco factor per fluke:

kfluke1 = 50.428 (3.36)

kfluke3 = 50.427 (3.37) 31

3.1.4 Conversion With the knowledge of the total voltage in equation 3.35, the spectra can be converted from voltage to frequency. This can be done by filling in the variables in equations 2.31 and 2.32 of section 2.2.2 about doppler-tuning. The masses are taken from Reference [29].

Figure 3.21: Example of 27Al spectrum before and after x-axis unit conversion.

Figure 3.21 shows a spectrum of 27Al before and after conversion. The width of the hyperfine spectrum is about 2500 MHz in frequency.

3.2 SATLAS - data fitting

With the spectrum in frequency units, the SATLAS program can be used for fitting and extracting the hyperfine parameters A and B in units of MHz. For this, a basemodel has to be created to represent the experimental data. Then the free parameters of the model are adjusted in the fitting procedure such that the model best matches the data. This is the basic premise of the fitting routine and is discussed in more detail in the following sections.

3.2.1 BaseModel The SATLAS package contains a HFSModel that is specifically used for the analysis of hyperfine spectra. It calculates the hyperfine splitting of 2 atomic levels based on the following free input parameters:

• I : nuclear spin of the aluminium isotope. This value depends on the isotope in question.

• J : electronic spins of the fine structure levels. In this case a transition from P3/2 to S1/2, so J = 3/2, 1/2 for the entire analysis. 32

• AB : hyperfine constants for lower and upper level of the transition. The values given as input are either rough estimates or taken from literature if available. During the fitting procedure, the A and B values will be optimized.

• centroid : The centroid or center of mass of the spectrum. This is the shift of the spectrum with respect to the transition frequency. It can be estimated as the offset of the largest peak from the zero point on the frequency axis.

With this input information and equations 2.22 and 2.25, the location of the transitions vi,j in frequency space can be determined based on the energy difference between 2 hyperfine levels Fi and Fj:

vi,j = v(I,Jupper,Fi,Aupper,Bupper) − v(I,Jlower,Fj,Alower,Blower) + centroid (3.38)

3.2.2 chi-squared fit routine The determination of the best agreement between experimental data and model function after fitting is based on the minimization of the cost function. The cost function represents how well the model function with its current set of parameters agrees with the data. A high value indicates more disagreement than a low value. This is why the cost function is minimized during fitting by altering the free parameters until the set of parameters is achieved for which the cost function has reached its lowest value. This set of parameters agrees best with the data and the fitting routine ends. The minimization of this function during fitting is done by altering the free parameters of the model until the best model and parameters are obtained that cannot agree more with the data and the fitting routine comes to an end. The SATLAS package uses the quickly converging Levenberg-Marquardt algorithm for the minimization of the cost function in the case of a sum-of-squares function. The cost function itself is the weighted χ2 formula:

N 2 X yi − f(xi) χ2 = (3.39) σ i=1 i

where yi is the experimental data, N is the amount of data points, f(xi) is the model value at frequency value x and σ is the uncertainty on each data point. Because this is i i √ a counting experiment, the uncertainty equals yi as a result of the underlying Poisson distribution.

The goodness of the fit is described by the reduced χ2 value, which is defined as the χ2 value per degree of freedom:

χ2 χ2 = (3.40) red N − #(freeparameters) where N indicates the amount of data points. A value of 1 means perfect agreement 2 between model and data. In this experiment, the χred values were consistently slightly higher than 1 which signifies an underestimation of the errors and that the fit has not fully captured the data. But only when the values are greatly higher than 1 a poor model fit is achieved. 33

Figure 3.22 shows an example of a fitting routine for 27Al, where the left figure shows the model and data before fitting and the right figure after fitting.

Figure 3.22: left: model and data before fitting. Right: model and data after fitting.

3.2.3 Weighted average After the fitting, the results can be gathered in a spreadsheet containing the values of a number of parameters for each measurement. The parameters include the hyperfine constants A and B and the centroid, but also other parameters that are discussed later on in subsection 3.2.4, such as FWHM and relative intensities of the peaks.

To get a final value of the parameters for each isotope, a weighted average is taken from the independent measurements. This average takes into account the error on each independent measurement and is given by equation:

PN xi i=1 σ2 hxi = i (3.41) weight PN 1 i=1 2 σi where the weights are the reciprocal of the error variance σ2 of each measurement. In this way measurements with large errors have less effect on the final result than more accurate measurements with small errors. The standard deviation σ is provided for each measurement by the fit program.

A final error that accompanies the weighted average is also needed. There is a statistical error that improves with increasing amount of measurements:

s 1 σ = (3.42) stat PN 1 i=1 2 σi and a scattering error [32]: v 2 uPN (xi−hxiweight) u i=1 σ2 σ = u i (3.43) scatt t PN 1 (N − 1) i=1 2 σi 34

The scattering error corresponds with the weighted standard deviation which is compara- ble to the computation of the weighted average. The final error is taken as the maximal of these errors. In this experiment, the scattering error was always the largest of the two.

3.2.4 Basemodel optimization The quality of the fitting process and the resulting parameters are largely dependent on the initial quality of the basemodel. Besides the input parameters that are used to calculate the position of the transitions, there are also other optional parameters that primarily influence the shape of the basemodel spectrum used for fitting. The optimal settings of these parameters were determined by variation until the best fitting results were obtained with respect to the data in this experiment. Some of these parameters could be estimated from theoretical considerations, others had to be determined by trial and error.

Lineshape and Linewidth One of the basemodel parameters is the lineshape and corresponding FWHM of the peaks. The lineshape can be Lorentzian, Gaussian or Voigt, which is a convolution of the previous two. To determine these parameters, an analysis of the linewidth can be performed. In section 2.2.2, the different contributions to the spectral linewidth of the atomic transition were discussed. There are 3 major contributions with their own shape and width:

• Doppler broadening: Equation 2.28 can be evaluated with the laser frequency taken from the labbook and the acceleration voltage from equation 3.35. With values taken from an example measurement of 27Al, a doppler width of 9.86 or approximately 10 MHz is obtained. This linewidth has a Gaussian shape.

• Natural linewidth: Equation 2.29 is evaluated by determining the lifetime τ of the excited state. This can be done by considering the Einstein coefficient A for 7 −1 spontaneous emission. For the transition 4S1/2 - 3P3/2, this value is 9.85 · 10 s . Since A = 1/τ, the lifetime is approximately 1 · 10−8 s [30]. The natural linewidth becomes about 16 MHz and has a Lorentzian shape.

• Power broadening: Equation 2.30 can be evaluated by finding the saturation laser power at which point increasing the power does not result in more photon emission because the maximal pumping power has been reached. Figure 3.23 shows a sat- uration curve taken from the labbook when the laser power was tested. It is not very clear due to the low amount of data points, but it can be deduced that the saturation power is around 2.1 mW. This estimation is enough since the goal is to have an estimated input FWHM value for the basemodel. The laser power used in the measurements was 1.9 mW, so the effect of power broadening is an increase of 38% compared to the original linewidth. This means that the Lorentzian natural linewidth is increased from 16 MHz to 22 MHz. 35

Figure 3.23: Saturation curve of influence of laser power on amount of detected fluores- cence photons. Figure taken from labbook.

Because the Gaussian contribution due to doppler broadening and lorentzian contribution due to power broadening are of the same order, the logical lineshape is the Voigt profile. The FWHM values can be taken as the list 10, 22 MHz representing the Gaussian and Lorentzian contribution respectively. The width of the Voigt profile is given by [31]: q 2 2 fV = 0.5346fL + 0.2166fL + fG (3.44) where fG and fL are the Gaussian and Lorentzian linewidths. This gives an estimated Voigt width fV of 26 MHz.

After the first analysis of the stable isotope 27Al, an average FWHM value of 173 MHz was obtained, which is by far larger than the theoretical expected value. One explanation is the presence of sidepeaks in the spectrum. Sidepeaks originate from inelastic scattering events between the ions and gas atoms in the CEC. It causes the ions to populate a different excited state and create spectrum peaks at some frequency offset left of the main peaks. This asymmetric broadening thus increases the linewidth. When corrected for the presence of sidepeaks, still a FWHM value of 105 MHz is reached. This discrepancy could be further explained by temperature fluctuations in the CEC, but I have found no tangible proof that could explain this high FWHM. Fortunately, the peaks of the hyperfine spectrum are all resolved and the the hyperfine constants A and B can be determined.

The analysis of 27Al shows that the Voigt lineshape leads to the best fit of the data. The 2 average χred value was 2.44 for the Gaussian shape, 1.61 for the Lorentzian shape and 1.42 for the Voigt shape. 36

Sidepeak parameters As mentioned above, sidepeaks can appear in the hyperfine spectrum and have an effect on the quality of the fit and the obtained fit parameters. It originates from following inelastic scattering events between the aluminium ions and the Na-vapor in the CEC:

Al+ + Na → Al+ − ∆E + Na∗ → Al+ − ∆E + Na + hv (3.45) Al + Na → Al − ∆E + Na∗ → Al − ∆E + Na + hv (3.46)

The first equation describes inelastic scattering before charge exchange has taken place, while the second equation occurs after charge exchange. In both cases, it is the fast Al-ions or Al-atoms that transfer momentum onto the Na-atoms and lose some of their velocity. This momentum transfer is indicated by the loss of energy ∆E. These inelastic collisions can happen to the same Al-atom several times. The probability of the same atom undergoing N successive collisions is given by Poisson’s law since the collisions are identical and independent:

xN P (N) = e−x (3.47) N! where x is a parameter dependent on the length and density of the Na-vapor [33]. For high temperatures and high vapor densities, the inelastic collisions will be more frequent.

An aluminium atom undergoing N such collisions will present itself in the spectrum as N satellite or ’side’ peaks at the low energy side of the main peak with an offset equal to N · ∆E. The SATLAS package allows to choose the amount of side peaks N which is not varied during fitting. It also allows to give an initial value to the relative intensity of the first side peak while the other side peak intensities are calculated using the Poisson distribution (see equation 3.47). The offset frequency of the first side peak can also be given and is varied during fitting.

Through a fit trial, the energy defect ∆E and thus amount of offset and the amount of 2 significant side peaks can be determined. Figure 3.24 shows the evolution of the χred when the fitting is performed with increasing amount of sidepeaks. It shows that a significant improvement is achieved up to an amount of 4 sidepeaks. This means there are 4 sidepeaks that influence the spectrum. In reality there are even more side peaks but due to the Poisson probability, their intensity drops sharply and do not significantly affect the spectrum. 37

2 Figure 3.24: Average χred values resulting from SATLAS fit procedure for different amounts of side peaks. A horizontal line is drawn through the value of 4 side peaks to guide the eye.

To determine the offset, the variable offset can be compared between fits of different side peak amounts. Figure 3.25 shows the offset for each measurement for varying number of sidepeaks fitted. With increasing amount of side peaks fitted, the offset is more and more uniform between measurements. Since the offset is not a random value, it should be the same for different measurements. This means that the offset value given by the fit procedure is only reliable starting from 4 sidepeaks where the scatter in offset has decreased. Table 3.4 shows the offset values calculated as a weighted average and the error. It shows that the offset value is between 70 and 71 MHz. 38

Figure 3.25: Offset in frequency space of sidepeak relative to main peak for measurements of 27Al. The offset is compared between fit procedures with varying amount of sidepeaks fitted.

Table 3.4: Offset values for various amounts of side peak fitting.

Sidepeaks (#) Offset (MHz) 1 -90 (22) 2 -72.8 (90) 3 -72.6 (85) 4 -70.6 (93) 5 -70.3 (74)

An offset of 70 MHz on the frequency spectrum of 27Al corresponds with an energy offset of 3.6 eV. This suggests that the inelastic scattering involves excitation to to 4P triplet located at this energy [34]. The energy offset is independent of mass and can be used to calculate the frequency offset for each isotopic mass under investigation using the formulas for frequency conversion: equations 2.31 and 2.32. Now the offset can be fixed for each mass during the fit process.

The importance of the sidepeak correction for the fit results can be seen in the values of the hyperfine constants. Figure 3.26 shows the result of the magnetic hyperfine constant Alower of the P3/2 atomic state for a different amount of sidepeaks considered during fitting. It matches better with the literature value with increasing amount of sidepeaks. Starting from 4 sidepeaks, the experimental value agrees within 1σ with literature. 39

Figure 3.26: Magnetic hyperfine constant Al of P3/2 level plotted against number of sidepeaks included in fit. Literature value taken from [35].

Other parameters Other settings for the basemodel were scale, background and racah intensities of the peaks. The scale was set for each measurement to the difference between the maximal and minimal amount of counts, while the background was taken as the minimal amount of counts.

The intensities of the peaks were given an equal initial value and were varied during fitting. There is the option to account for saturation in the spectra. Since none of the experimental spectra showed saturation, there was no difference in the fitting result when saturation was on or off.

3.2.5 Ratio Au /Al In order to reduce the amount of variables and in doing so reduce the error on obtained fit variables, some parameters can be fixed to a known value. One of those parameters is the ratio of the hyperfine constant Au of the S1/2 level to the hyperfine constant Al of the P3/2 level. Along an isotopic chain, this ratio has a constant value. This can be seen from equation 2.23: the values of µ and I are the same for Au and Al, so the ratio is determined by the values of the hyperfine magnetic field B(0) and the electronic spin J of the P3/2 and S1/2 atomic state. Between different isotopes the ratio of J remains the same, since the same transition is investigated in each isotope. Also the value of B(0) is the same because only the neutron number changes between isotopes: the electron cloud that creates the magnetic field at the nucleus stays the same.

Table 3.5 shows the ratios for the different isotopes under investigation in this experiment. The weighted average of these values is taken as the fixed ratio value for the fit procedure. Table 3.5: Ratio of hyperfine constant A of each isotope. Nuclear spin of each isotope is added between brackets.

Isotope (I) A-ratio 26Al (5) 4.486 (14) 27 5 Al ( 2 ) 4.572 (5) 28Al (3) 4.567 (5) 29 5 Al ( 2 ) 4.573 (2) 30Al (3) 4.562 (12) 31 5 Al ( 2 ) 4.556 (14) 4.571 (10)

The fixed ratio fitting was tested for 26Al because, as can be seen from Table 3.5, only this isotopes does not agree with the average A-ratio. With fixed ratio, there was a reduction of 0.06 in the error of Al. This reduction is not observed for Au. On the other hand the Bl value moves further away from the literature value and does not correspond with it anymore within a 1σ error. It was decided to not constrain the A-ratio since the improvement in error on Al does not weigh up against the negative effect on Bl and the non-effect on Au.

As a side note it must be said that the constant A-ratio is only valid when the hyperfine anomaly is negligibly small. The hyperfine anomaly takes into account the distribution of the nuclear charge and the extended nuclear magnetisation over the nuclear volume [36]. There is a difference between the point-like and the actual magnetic hyperfine interaction that manifests itself in the value of the hyperfine constant A. This hyperfine anomaly is different for different isotopes, which causes the A-ratio to not remain constant along the isotopic chain. According to the summary of known hyperfine anomalies from 2011 in reference [36], the anomaly for aluminium is not known. Looking at Table 3.5, it is expected that the hyperfine anomaly is indeed negligibly small.

4 Results

This chapter contains the results regarding the hyperfine constants obtained by the fitting procedure described in the previous chapter. The magnetic dipole moment µ and electric quadrupole moment Q can be calculated through the literature values of the reference isotope 27Al and the data on the hyperfine constants obtained in this experiment. The accuracy of the determination of nuclear moments with the COLLAPS method is inferred from the comparison with methods used in literature. Finally, the nuclear moments of 29Al and 30Al are compared with theoretical calculations to determine their ground state

40 41

configuration and their position with respect to the island of inversion.

4.1 Hyperfine Constants

Table 4.6 contains the hyperfine constants of the measured isotopes in this experiment as a result of the fit process. The values and errors are calculated using equations 3.41 and 3.43. Literature values are found only for 27Al and 26Al.

2 2 2 Table 4.6: Summary of hyperfine constants A and B of the 3s 3p P3/2 level and the 3s 4s 2 S1/2 level of the aluminium isotopes investigated.

Isotope 26Al 27Al 28Al 29Al 30Al 31Al

Al (P3/2) (MHz) 36.75(11) 94.33(5) 70.14(13) 94.90(11) 65.21(35) 99.20(81) literature 36.53(48) [37] 94.25(4) [35] - - - -

Au (S1/2) (MHz) 164.84(27) 431.24(10) 320.36(41) 433.90(24) 297.5(11) 451.99(80) literature [38] - 421(15) - - - -

Bl (P3/2) (MHz) 35.4(18) 18.08(33) 17.48(90) 17.85(83) 15.0(30) 21.0(30) literature 33.2(37) [37] 18.76(25) [35] - - - -

For where they are available, the literature values agree with the experimental values 27 except for the Bl value of Al. The experimental result underestimates this value. The 27 hyperfine constant Au of the S1/2 state of Al determined in this experiment shows improvement in accuracy over the literature value with a large decrease in uncertainty 26 from 15 to 0.1 MHz. Also the experimental errors on Al and Bl of Al are smaller but of the same order as the literature errors.

4.2 Nuclear Moments

With the knowledge of the hyperfine constants in Table 4.6, the nuclear moments can be calculated. Additionally, the nuclear moments of one of the isotopes is also required to serve as a reference. In this experiment the nuclear moments of the stable 27Al isotope is used from literature and we can consider the ratio between the magnetic moment of a certain isotope and the reference isotope:

IJA µ B(0) = I J A (4.48) µref ref ref ref B(0)ref

The ratio resulted from using equation 2.23. The electronic spins J and Jref cancel if both magnetic moments are calculated using the hyperfine constant A of the same atomic state: P3/2 or S1/2. The magnetic fields B(0) and B(0)ref are equal and also cancel since the electron cloud is the same for each isotope. Now an equation is achieved to determine µ that is only dependent on the hyperfine constants (experiment), the nuclear spins (literature) and the reference magnetic moment (literature): AI µ = µref (4.49) Aref Iref 42

For the calculation of the magnetic moments with equation 4.49 always the Au hyperfine constant of state S1/2 is used in this experiment. Its value is significantly higher than that of Al of state P3/2 and since the uncertainties on both constants are of the same order of magnitude, the relative uncertainty is lower for Au.

The same reasoning can be used on the electric quadrupole moment using equation 2.26. This gives the ratio:

B Q eVzz = B (4.50) Qref ref eref Vzz/ref

The charges e and eref are both the same and equal to the elementary charge. The electric field gradients Vzz and Vzz/ref are both generated by the electron cloud. Since the electron cloud is the same for different isotopes, these values cancel. After simplification, equation 4.50 leads to the following equation: B Q = Qref (4.51) Bref

For the calculation of the quadrupole moments, the Bl hyperfine constant of state P3/2 is used in this experiment.

4.2.1 Magnetic Dipole Moment Using equation 4.49, the magnetic dipole moments of the isotopes can be calculated. Table 4.7 lists these values together with literature values for comparison.

Table 4.7: Experimental and literature values of the Magnetic dipole moments of the aluminium isotopes in units of nuclear magneton (µN ). Literature values taken from [40].

Isotope µexp [µN ] µliterature [µN ] 26Al 2.784(5) 2.804(4) 27Al 3.641(1) 3.6415069(7) 28Al 3.246(4) 3.242(5) 29Al 3.664(2) - 30Al 3.014(12) 3.010(7) 31Al 3.817(7) 3.830(5)

The values in table 4.7 are visually represented on Figure 4.27. The overlap with literature is confirmed for each aluminium isotope, except for 26Al and 31Al. The experimental values of 26Al and 31Al fall outside the errorbars of the literature value with a margin of respectively 0.011 and 0.001. The disagreement of 31Al could be explained by the statistics and quality of the measurements, which is explained further in section 4.2.2. The value of 29Al cannot be compared because of the non-existence of a literature value. 43

Figure 4.27: Plot of magnetic moments in table 4.7 in sequence from A=26 to A=31. Error bars are included, but too small to be visible.

Figure 4.27 shows a distinction between the magnetic moment values of the even A and odd A isotopes. The odd A isotopes contain an uneven amount of protons (13) and always an even amount of neutrons. The similarity of the magnetic moment for the odd isotopes illustrates the sensitivity of the magnetic moment to the orbital occupied by the last unpaired nucleon. In this case the proton occupies the 1d5/2 orbital in the nuclear shell model. In fact it is the g-factors of the nuclei that show the similarity, but since the nuclear spin of 27Al, 29Al and 31Al are all equal to 5/2, the magnetic moments are the same as well. The g-factor is obtained by simply dividing the magnetic moment by the nuclear spin I.

Figure 4.28 compares the experimental g-factor values for the odd A isotopes with the g-factor for an unpaired nucleon on the d5/2 orbit calculated using the Schmidt moments given by equation 1.9. The first calculation was done using the free g-factors of equation 1.7. The second calculation was done using the effective g-factors. These g-factors take into account the presence of a nuclear medium and correct for meson exchange interactions and core polarization effects [10]. In this region of the nuclear chart, the values used for l l s s a proton are: geff = 1.1 · gfree, geff = 0.7 · gfree [39]. As can be seen on Figure 4.28, the experimental values lie closer to the effective g-factor than the free g-factor. The still remaining offset from the effective g-factor is further explained by the presence of configuration mixing and the fact that the single particle picture is in this case too simplistic to fully predict the correct g-factor. 44

Figure 4.28: Experimental g-factors of the odd A isotopes in sequence 27-29-31. Error bars are too small to be visible on the figure. Also plotted are the Schmidt g-factors for a nucleus with unpaired nucleon in d5/2 orbital. The red line represents the Schmidt value calculated using the free g-factors, the blue line represents the effective g-factors.

The even A isotopes have an unpaired proton as well as an unpaired neutron. The g-factor is influenced by interaction between the 2 nucleons and the orbits they occupy and is thus not equal for each even A isotope.

4.2.2 Electric Quadrupole Moment Using equation 4.51, the electric quadrupole moments of the isotopes can be calculated. Table 4.8 lists these values together with literature values for comparison.

Table 4.8: Electric quadrupole moments of aluminium isotopes in units of barns (b). Literature values taken from [40].

Isotope Qexp [b] Qliterature [b] 26Al 0.287(16) 0.265(32) 27Al 0.1466(39) 0.1466(10) 28Al 0.142(8) 0.175(14) 29Al 0.145(7) - 30Al 0.121(24) - 31Al 0.170(23) 0.134(2)

Figure 4.29 gives a graphical representation of the values in table 4.8. The experimental values correspond with literature within errorbars, except for 28Al and 31Al. For 29Al and 30Al there are no literature values available. 45

Figure 4.29: Plot of quadrupole moments in table 4.8 in sequence from A=26 to A=31.

The uncertainties on the electric quadrupole moment are larger than the uncertainties on the magnetic dipole moment. An explanation for this can be found in the magnitude of the hyperfine splitting caused by the dipole and quadrupole interaction. The quadrupole interaction has a smaller effect on the hyperfine structure of aluminium. Figure 2.16 shows that the coefficients of the B constants are consistently smaller than those of the A constants. This makes the determination of B less accurate. The larger absolute uncertainty on B together with the fact that B is smaller in value than A translates in a large relative error. This relative error on B causes the larger error on Q compared with µ.

28Al and 31Al show a significant deviation both of 0.011 in size from the literature uncer- tainty interval. During the analysis of 28Al, there was no evidence of a possible cause for this deviation. 31Al has only 3 independent measurements to use in the analysis. Also these 3 spectra do show some amount of scatter in the peaks as can be seen on Figure 4.30. Due to the low available amount of spectra, there was no luxury to dismiss these measurements. This could be the reason of the deviation in quadrupole moment and magnetic dipole moment for 31Al. 46

Figure 4.30: The 3 measurements of 31Al used in the analysis.

4.3 Comparison with Literature Methods

The accuracy of the results achieved in this experiment can be tested by comparing the COLLAPS method with other methods used to determine nuclear moments in the literature. The literature values for the nuclear moments listed in Table 4.7 and Table 4.8 were determined using different methods:

The moments of 26Al were measured by atomic beam laser spectroscopy (ABLS), similar to this experiment. The magnetic moment came from an article by Cooper et al. [41] through investigation of the P1/2-D3/2 transition, while the quadrupole moment came from another article by Levins J. et al. [37] through measurement of the P3/2-D3/2 transition. The magnetic moment determined in this experiment has an uncertainty of the same order of magnitude as in literature. The value on the other hand deviates from literature by a small amount as seen in section 4.2.1. This deviation could be explained by the transition mixing due to the proximity of the D5/2 to the D3/2 level, which would make the literature value less precise. Another explanation could lie in the higher FWHM of the peaks obtained in this experiment compared with the literature experiment (110 MHz vs 75 MHz). A high FWHM results in a less precise determination of the position of the peaks in the spectrum. This would indicate that the experimental value is further from reality than the literature. Further investigation would have to be made to determine which value is closer to reality. The experimental quadrupole moment of 26Al agrees with literature and has a better accuracy. This could be due to the larger amount of measurements compared to the literature experiment (17 vs 11).

The stable isotope 27Al has remarkable lower uncertainties than the other isotopes. The techniques used to determine the nuclear moments involve the investigation of aluminium embedded in molecules and is described in articles by Kello V. et al. [42] and Epperlein B. et al. [43]. The magnetic moment ([43]) is determined by applying NMR to Al- 47

ions dissolved in water to find the Larmor frequency. From the ratio with the Larmor frequency of 2H, the magnetic moment can be determined. The quadrupole moment ([42]) is determined by direct calculation of the electric field gradient of Al-ions embedded in molecules such as AlF and AlCl. Because 27Al is stable, these techniques can be used and higher accuracies can be achieved.

The nuclear moments of 28Al, 30Al and 31Al were determined using nuclear magnetic/quadrupole resonance with β-asymmetry detection (β-NMR/NQR). In a β-NMR experiment, a spin- polarized atomic beam is exposed to an external magnetic field which induces a Zeeman- splitting of the magnetic substates. The spin-polarized beam has an asymmetric β-decay pattern. The magnetic dipole moment is determined by matching the frequency of an applied rf-field to the Larmor frequency. This will cause equalized population of the substates and destruction of the asymmetry which is expressed by a peak or dip in the spectrum. The accuracy of the technique is, like for COLLAPS, based on how well the position of the peak in frequency space can be determined. In a β-NQR experiment, the presence of an electric field gradient together with a mag- netic field is required which causes a non-equidistant splitting of states. It requires the determination of the quadrupole coupling frequency as well as the use of the Larmor frequency determined by β-NMR. Because of the non-equidistant splitting, multiple rf- frequencies have to be applied simultaneously to maximize the destruction of asymmetry [45]. Finally, a mismatch in alignment between the magnetic field axis and the EFG axis can cause the resonance frequency that is found to deviate from the actual frequency which leads to a shift in the quadrupole moment determination. These effects cause the quadrupole moment extracted in this way to be less accurate than the magnetic moment determined through β-NMR. The experimental nuclear moments of 28Al and 30Al have comparable uncertainties to those determined by the NMR-techniques of literature. The moments of 31Al are less accurately determined in this experiment. This might be caused by the low statistics with only 3 measurements.

In conclusion, the results of this experiment by the COLLAPS set-up achieve similar accuracies as the methods used in literature. Results from other experiments using the COLLAPS set-up ([44]) to determine nuclear moments show uncertainties for the mag- netic dipole moment comparable to literature values obtained by other ABLS methods and β-NMR. If we look at equation 2.23, we see that there is a general improvement in relative error on the hyperfine constant A for heavier nuclei. Heavier nuclei have larger magnetic fields generated by a larger electron cloud which results in a larger value for A. It is expected to have more accurate magnetic moments for heavier nuclei than lighter nu- clei. Hyperfine parameter B does not have this mass dependence and is generally smaller than A. The quadrupole moment in other studies show uncertainties of the same magni- tude as in this study. From this, it can be concluded that the COLLAPS method is suited to accurately deter- mine nuclear moments that can compete with results from other methods. 48

4.4 New Results: 29Al and 30Al

The experiment has led to some new results, namely a value for the magnetic dipole moment of 29Al and quadrupole moment for 29Al and 30Al. Although these values cannot be compared to literature values, they can be compared to theory. Using large-scale shell model calculations, theoretical values for the nuclear moments can be found. Calculations are performed using the USD interaction discussed in section 1.3.1 and the SDPF-M interaction. The USD interaction valence space limits itself to the sd-shell, while SDPF-M also includes part of the pf-shell (f7/2 and p3/2) in the valence space available for neutrons. For computational purposes, there is a limitation of maximal 2 neutron excitations to the pf-shell. Both interactions were calculated using the KSHELL program implementing a full diagonalisation, similar to the ANTOINE code. Comparison with these two interactions provides information about the ground state configuration of our isotopes and allows us to determine their position with respect to the island of inversion.

Figure 4.31 shows the experimental magnetic dipole moments compared to theoretical values. The free nucleon g-factors of equation 1.7 were used to determine the theoretical values. Calculated values for 27Al and 31Al are also included for comparison purposes.

Figure 4.31: Comparison between experimental and calculated magnetic moments of 27, 29, 30, 31-Al. Calculations performed with USD and SDPF-M interaction.

From the figure, it can be seen that the newly determined experimental µ-value of 29Al corresponds with the theoretically predicted value. The correspondence between USD and SDPF-M calculated values shows that the inclusion of the pf-shell makes no difference for the magnetic moment. This means that there are no significant excitations by neutrons from the sd to pf orbitals in the ground state configuration.

Besides the magnetic moment, the quadrupole moment is also and even more sensitive to the the presence of intruder states across the shell gap at N = 20. This is because of the link between nuclear deformation and intruder states. The presence of intruder 49 states creates an extra deformation in the nucleus on top of the expected deformation for the normal sd-shell ground state. Since the quadrupole moment is a measure of nuclear deformation, it is therefore the best test to determine the amount of intruder state mixing. [10] Figure 4.32 shows the experimental quadrupole moments compared to theoretical values. Effective charges eπ = 1.1e and ev = 0.5e were used in the calculation of the theoretical values which are most suited to use in sd-shell calculations [45].

Figure 4.32: Comparison between experimental and calculated quadrupole moments of 27, 29, 30, 31, 33-Al. Calculations performed with USD and SDPF-M interaction. Experi- mental and calculated Quadrupole moments of 33Al are taken from an article by Heylen, H. et al. ([46])

A significant difference between USD and SDPF-M quadrupole moments indicates the presence of intruder states and can be used to determine the amount of mixing between sd and pf configurations in the ground state of the aluminium isotopes [45]. 27Al is already confirmed to be normal sd-configuration, while 31Al is mainly sd-shell configuration with possible small admixture of pf-shell excitation as stated by previous studies (De Rydt, M. et al.: [45]). For 31Al, the literature value is added because of the deviation of the experimental value. The uncertainty of the literature value is smaller and a more accurate comparison with theory can be made than with the experimental value. The literature value tells us that the 31Al ground state consists of normal sd-shell configuration.

The experimental quadrupole moment of 33Al is added on Figure 4.32 and is taken from an article by Heylen, H. et al. ([46]). In this case, there is a clear difference between the Q predicted by USD and by SDPF-M interaction. 33Al has a fully filled neutron shell (N = 20), so contributions to the quadrupole moment have to come from the proton config- uration. The result is a low Q predicted by USD. SDPF-M on the other hand includes neutron excitations to the pf-shell. These neutrons can contribute to the quadrupole moment resulting in a higher predicted Q. The experimental value lies between the pre- dictions, somewhat closer to the SDPF-M value. This indicates the presence of intruder states and a mixing of sd- and pf-shell configurations in the g.s. of 33Al. 50

In comparison with the case for 31Al and 33Al, it can be clearly seen on Figure 4.32 that 29Al and 30Al both show no sign of intruder mixing and can be identified as purely normal sd-configuration. With the information that 32Al also shows a pure sd-shell configuration [45], it can be said that the onset of intruder state mixing in aluminium isotopes starts at 33Al.

Table 4.9 contains an overview of the experimental and calculated values used in Figures 4.31 and 4.32.

Table 4.9: Overview of experimental and calculated nuclear moments for a sample of aluminium isotopes. Experimental and calculated Quadrupole moments of 33Al are added and taken from an article by Heylen, H. et al. ([46]). Magnetic moments are calculated with free nucleon g-factors. Quadrupole moments are calculated with effective charges eπ = 1.1e and ev = 0.5e.

Isotope 27Al 29Al 30Al 31Al 33Al

µ [µN ] experiment 3.641(1) 3.664(2) 3.014(12) 3.817(7) - USD 3.560 3.711 3.033 3.810 - SDPF-M 3.604 3.733 3.103 3.798 - Q [mb] experiment 147(4) 145(7) 121(24) 170(23) 141(3) [46] USD 144 142 117 142 100 SDPF-M 146 140 113 136 154 5 Conclusion

The hyperfine structure of aluminium isotopes from N=13 to N=18 was investigated by means of the COLLAPS beam line. In contrast to previous studies where the transition from the P-doublet to the D-doublet was used ([41], [37]), the transition P3/2 - S1/2 was used to create the hyperfine spectrum in this experiment. This eliminates transition mixing between the D-doublets.

The achieved results for the nuclear moments mostly match with literature values within errorbars. This experiment also led to the determination of the previously unknown nuclear moments of 29Al and unknown quadrupole moment of 30Al. 27 The magnetic hyperfine constant A of S1/2 for Al shows an improvement in accuracy by 2 orders of magnitude compared to literature and should be used in future references. Also the quadrupole moment of 26Al has been more accurately determined with an uncertainty half as small as determined in previous experiments.

The determined nuclear moments were compared to theoretical predictions of large scale shell model calculations. The USD and SDPF-M interaction were used for the calculations to investigate the presence of intruder state mixing in the ground states of the aluminium isotopes. The newly obtained values for the nuclear moments of 29Al and the quadrupole moment of 30Al agreed with theory and showed no presence of intruder states in the ground state. The agreement with theory places trust in the correctness of the newly found moments. It also confirms the expectation that these isotopes are placed outside the island of inversion around N=20 and Z=10-12. The aluminium isotopes lie at the border between the island of inversion and the normal non-deformed nuclei. These regions often contain the presence of transitional isotopes that show a g.s. mixture of normal and deformed configuration, such as 33Al. Investigation of these regions help to better understand how the transition to the region of deformed nuclei occurs.

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