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FACULTEIT WETENSCHAPPEN

Nuclear Moments alongaluminium isotopic chainusing Collinear LaserSpectroscopy

Pieter SMOLDERS

Promotor: Prof. dr. G. NeyensBegeleider: W. Gins

Proefschrift ingediend tot het

behalen van de graad van

Master of Science in de Fysica

Academiejaar 2016-2017

Dankwoord

Aangezien deze thesis alleen maar tot stand kon komen dankzij het onderzoek dat inteamverband is uitgevoerd en de hulp die ik gekregen heb van de mensen die achter ditonderzoek stonden, wil ik eerst hier mijn dankbetuigingen geven.Eerst en vooral wil ik mijn promoter in Leuven, Prof. Dr. Gerda Neyens, bedanken ommij 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 teobserveren hoe een wetenschappelijk experiment op hoog niveau wordt uitgevoerd.Hierbij wil ik ook mijn begeleider Wouter Gins bedanken die mij wegwijs heeft gemaaktin Geneve en mij geholpen heeft met al de voorschriften om binnen te geraken. Daarnaastbedank ik hem ook voor de begeleiding in Leuven en voor alle technische hulp met hetanalyse programma SATLAS.

Many thanks also to all the phd-students of the COLLAPS collaboration that were presentduring the measurements. Thanks also to Livio who was also for the first time in CERNand with whom I spent some time going to all the mandatory safety courses. Specialthanks go out to the people of the morning shift whom I had the chance to get to knowbetter. This includes Ronald who had interesting views on electrical sockets from aroundthe world. Also Xu that accompanied me during the morning commute from our residenceand who also helped me in Leuven with the shell model calculations. And let’s not forgetthe 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 andgive good advice during the progress meetings and helped me complete the analysis andthis thesis.

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Abstract

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

Research into the nuclear structure and the shell configuration that the neutrons andprotons 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, anddetermines the effective interactions between neutron and proton. Over the years, therehave 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 theinteractions between the nucleons. A better understanding of the strong force can thuslead to better shell models.

It is predicted there are approximately 6000 different nuclei in the universe. Only afraction of this amount has been investigated and most are located close to the valley ofstability on the nuclear chart. This means the shell models that are being used at themoment are mainly based on the behaviour of the strong force in stable nuclei. DuringResearch into the so called exotic nuclei, deviations between experiment and shell modelpredictions have been discovered. This means that the interactions between nucleons aredifferent for these nuclei than for stable nuclei. To elucidate the behaviour of the strongforce in all regions of the nuclear chart and to create better effective interactions to beused 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 aroundN=20. However the exact borders of this island has not entirely been determined. Thenuclei located inside the island exhibit a different shell configuration than expected bystandard shell models for nuclei in that area of the nuclear chart. Normally, an energygap is present between N=20 and N=21. This energy gap identifies the transition fromone shell to another. The neutron or proton number at which a shell is filled is called amagic number. The standard shell model thus predicts a magic number at N=20. Thenuclei inside the island of inversion on the other hand show a strongly reduced magicity ofN=20 and therefore have a smaller energy gap. As a result these nuclei exhibit a groundstate 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

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31Al by using Collinear Laser Spectroscopy (COLLAPS set-up). The magnetic dipolemoment and electric quadrupole moment are determined. These moments are sensitiveto the structure of the nucleus and can be used to check the validity of the normal shellmodel thereby seeing if there is any deviation in strong force behaviour. Specific attentionwill be given to 29Al and 30Al since both moments (magnetic and electric) are missing for29Al and the electric quadrupole moment is missing for 30Al. Because aluminium has 13protons and lies at the edge of the island of inversion, both these isotopes will be checkedto exhibit the same deviation in strong force behaviour as isotopes belonging to islandof inversion. Expectation is that they exhibit a normal ground state configuration since31Al has a normal ground state, as confirmed in previous research.

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

Samenvatting

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

Onderzoek naar de structuur van kernen en de schilconfiguratie waarin de neutronen enprotonen zich bevinden kan informatie verschaffen over een van de fundamentele krachten,namelijk de sterke kracht. Deze kracht zorgt voor de binding van de kerndeeltjes tot eengeheel, de kern, en bepaalt de effectieve interactie tussen neutron en proton. Over de jarenheen zijn er verschillende schilmodellen opgesteld die voorspellingen kunnen doen over deeigenschappen van kernen in verschillende regio’s op de nucleaire map. Deze modellenzijn gebaseerd op de effectieve interacties tussen de kerndeeltjes. Een beter inzicht in desterke kracht kan dus leiden tot betere schilmodellen.

Er is voorspeld dat er 6000 verschillende kernen bestaan in het universum. Hiervan isnog maar een fractie daadwerkelijk onderzocht en deze kernen bevinden zich voornamelijkdicht bij de vallei van stabiliteit op de nucleaire map. Dit betekent dat de schilmodellenvooral 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 wilzeggen dat de interacties tussen de kerndeeltjes voor deze kernen anders zijn dan in devallei van stabiliteit. Om het gedrag van de sterke kracht in alle gebieden van de nucleairemap te ontsluieren en om betere interacties voor het schilmodel te produceren, is de studievan deze exotische kernen van groot belang.

Een voorbeeld van dit afwijkend gedrag is het eiland van inversie. Een van deze eilandenbevindt zich op de nucleaire map van Z=10 tot Z=12 en rond N=20, hoewel de exactegrenzen van dit gebied nog niet volledig bepaald zijn. De kernen in deze regio vertoneneen heel andere schilconfiguratie dan voorspeld via het standaard schilmodel voor kernenin 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 neutronof 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 gelegenin 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 neutronexcitatie tot de volgende schil.

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

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een reeks aluminium-isotopen bestuderen. Van 26Al tot31Al werd het magnetisch dipoolmoment (µ) en elektrisch quadrupool moment (Q) bepaald. Deze momenten zijn gevoeligaan de structuur van de kern en kunnen gebruikt worden om de validiteit van het schilmodelte controleren en te kijken naar mogelijke afwijkingen van de sterke kracht. Specifiek van29Al zijn zowel het magnetisch dipool als electrisch quadrupool moment nog niet eerderbepaald, terwijl van 30Al het quadrupool moment niet gekend is. Aangezien aluminiummet zijn 13 protonen aan de rand van het eiland van inversie ligt, wordt er nagegaanof deze 2 isotopen dezelfde afwijking vertonen. De verwachting is dat deze isotopen eennormale grondtoestand vertonen aangezien voor 31Al al eerder bepaald werd dat het eengewone 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 numberN=20 en dus niet tot het eiland van inversie behoren.

Contents

Dankwoord i

Abstract ii

Samenvatting iv

0 Introduction 1

1 Theory 21.1 Nuclear Shell model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2

1.1.1 Shell structure evolution in exotic nuclei . . . . . . . . . . . . . . . 51.1.2 Island of inversion . . . . . . . . . . . . . . . . . . . . . . . . . . . 6

1.2 Nuclear electromagnetic moments . . . . . . . . . . . . . . . . . . . . . . . 81.2.1 Magnetic dipole moment . . . . . . . . . . . . . . . . . . . . . . . . 81.2.2 Electric quadrupole moment . . . . . . . . . . . . . . . . . . . . . . 9

1.3 Large Scale Shell Model calculations . . . . . . . . . . . . . . . . . . . . . 111.3.1 USD interaction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11

2 Experiment 142.1 ISOLDE - isotope production . . . . . . . . . . . . . . . . . . . . . . . . . 142.2 Collinear Laser Spectroscopy . . . . . . . . . . . . . . . . . . . . . . . . . . 17

2.2.1 Hyperfine structure . . . . . . . . . . . . . . . . . . . . . . . . . . . 172.2.2 COLLAPS . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19

2.3 Data . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 242.3.1 Files . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 242.3.2 P3/2 → S1/2 Transition . . . . . . . . . . . . . . . . . . . . . . . . 25

3 Analysis 283.1 Voltage to Frequency conversion . . . . . . . . . . . . . . . . . . . . . . . . 28

3.1.1 ISCOOL Voltage . . . . . . . . . . . . . . . . . . . . . . . . . . . . 283.1.2 Fluke Voltage . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 293.1.3 Scanning voltage . . . . . . . . . . . . . . . . . . . . . . . . . . . . 303.1.4 Conversion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31

3.2 SATLAS - data fitting . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 313.2.1 BaseModel . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 313.2.2 chi-squared fit routine . . . . . . . . . . . . . . . . . . . . . . . . . 323.2.3 Weighted average . . . . . . . . . . . . . . . . . . . . . . . . . . . . 333.2.4 Basemodel optimization . . . . . . . . . . . . . . . . . . . . . . . . 34

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

4 Results 404.1 Hyperfine Constants . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 414.2 Nuclear Moments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41

4.2.1 Magnetic Dipole Moment . . . . . . . . . . . . . . . . . . . . . . . 424.2.2 Electric Quadrupole Moment . . . . . . . . . . . . . . . . . . . . . 44

4.3 Comparison with Literature Methods . . . . . . . . . . . . . . . . . . . . . 464.4 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 electricquadrupole 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 modelcalculations and can be subsequently identified as being part of the island of inversion ornot.

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

Chapter 1 describes the nuclear shell model framework. This includes an introductionto the nuclear shell model and how it changes when handling exotic nuclei. An exampleof 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 adescription of the large scale shell model calculations appropriate for the isotopes underinvestigation is given.

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

Chapter 3 discusses the analysis of the raw data and how the nuclear moments are obtainedfrom 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 withliterature and shell model predictions. This chapter contains tables with hyperfine con-stants, magnetic dipole moments and electric quadrupole moments. The new nuclearmoments for 29Al and 30Al are compared with USD and SDPF-M shell model calculationsto investigate their ground state wave function.

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

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1 Theory

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

1.1 Nuclear Shell model

The nuclear shell model came into existence by applying the principles of atomic theoryto the nuclear realm. In atoms, the electrons moving around the nucleus occupy certainorbits at certain distances from the center of the atom. Each orbit can contain a certainamount of electrons based on the quantum numbers of that particular orbit and thePauli exclusion principle, which states that no 2 fermions can have the same quantumnumbers. The atomic properties are then determined by the electrons in the last unfilledshell, while the filled shells are considered an inert core. In terms of the nucleus, thistranslates to protons and neutrons moving in certain proton and neutron shells. Eachshell contains a certain amount of nucleons depending on the quantum numbers of thenuclear states contained in that shell and also the Pauli exclusion principle that limits theamount to one nucleon for each different set of quantum numbers. The nuclear propertiesare determined primarily by the unpaired nucleon in the last unfilled shell. This analogyformed 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 ofneutrons [2]. At certain neutron numbers a sudden increase in charge radius is observedas seen in Fig. 1.1, much like the increase in atomic radius when increasing the amountof electrons shown in Fig. 1.2.

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Figure 1.1: Change in nuclear charge radius as a function of neutron number N. At themagic 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 cleardiscontinuities 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 numbersat which there is a discontinuity of the charge radius. The nucleons inside these shellsare expected to have a stronger binding energy which explains why they are spatiallycloser to each other compared to a nucleon in a subsequent shell, hence the increase incharge radius. The proton and neutron numbers at which a shell is filled are called magicnumbers 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 appropriatenuclear potential had to be chosen. The main characteristics being the attractive centralpotential starting at intermediate distances and the repulsion at short distance. TheWoods-Saxon potential (equation 1.1) fulfills these requirements [2]:

V (r) =−V0

1 + exp[ (r−R)a

](1.1)

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 energystates of the nucleons. The interaction between spin s and angular momentum l splits the

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energy levels and labels them with a new quantum number j = l+ s and a degeneracy of2j+ 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 aWoods-Saxon potential.

Figure 1.3: Energy levels and their degeneracy obtained with a Harmonic Oscillatorpotential (left). Spin-orbit correction to energy levels (right). The shell structure isvisible 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 ofa nucleon inside a nucleus governed by a mean potential [4]:

H =A∑i=1

( p2i2mi

+ Ui(ri))

(1.2)

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 othernucleons.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 correctmagic numbers according to experiment are obtained. These levels represent the protonand neutron shells alike, however the proton shells will be shifted to higher energy due tothe charge repulsion between protons. Each level is labeled with the quantum numbersn, l and j. Making the analogy with atomic theory again, the energy levels labeled byquantum number j can be considered as orbits on which the nucleons move about in the

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nucleus. The energy of these single particle levels are also referred to as the Single ParticleEnergies 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 nucleuscorresponds to filling the single particle levels with nucleons starting at the lowest leveluntil A nucleons are exhausted. The properties of the nucleus are then determined by theSPE in which the last unpaired nucleon is located [2]. For example, 41Ca has only oneneutron beyond doubly closed shell (Z=20, N=20). The first level above this closed shellis f7/2, so the nucleus will have a nuclear spin I equal to the total angular momentum j=7

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of the last unpaired nucleon. Doubly closed 40Ca will have a ground state spin of 0 sinceall nucleons are paired. To create an excited state in this nucleus, a pair of nucleons hasto be broken to excite one of them to the next shell. So, a shell gap has to be crossedand the properties of the excited state is then determined by the coupling of the 2 oddnucleons.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 nucleonsto interact, this method starts to break down. To see why, a closer look has to made atthe nuclear Hamiltonian.

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

H =∑

εaa†αaα +

1

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∑< αβ|V |γδ > a†αa

†βaδaγ (1.3)

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. εa 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 partof equation 1.3 removes 2 particles from the states γ and δ and adds 2 particles in thestates α and β. The matrix element governs this interaction. This additional part to thenuclear Hamiltonian causes changes in the SPE’s when moving away from stable isotopesand towards the drip lines on the nuclear chart. This is why it is important to considerthe 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 bediscussed in more detail.

Monopole interaction

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

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Figure 1.4: Tensor interaction between neutron in orbit j′ and proton in orbit j. Picturetaken from reference [6]

A neutron in orbit j′ interacts with a proton in orbit j via the monopole part of theresidual nuclear interaction potential Vm. This causes an energy shift of orbit j equal to

∆ε(j) =1

2(V T=0

m (j, j′) + V T=1m (j, j′))n(j′) (1.4)

where n(j′) is the occupation number of neutron orbit j′ and T refers to the differentisospin states the proton-neutron system can have.[6]. Formula 1.4 shows that a largeroccupation of j′ results in a stronger tensor interaction and thus energy shift of the orbitj. An explanation of determining in which direction the level is shifted lies in the relativeangular momentum of the two nucleons. If a neutron in orbit j′< = l′− 1/2 interacts witha proton in orbit j> = l+ 1/2, then the large relative momentum causes the spatial wavefunction of the 2-nucleon system to be narrowly distributed. On the other hand, if orbitj′< interacts with orbit j<, then the relative momentum will be small due to same rotationdirection and the wave function will be spatially stretched. The first case describes anattractive interaction and thus when a nucleon in orbit j′< interacts with nucleon in orbitj> or j′> interacts with j< the energy level of orbit j will be shifted down. The secondcase corresponds with a repulsive interaction and will result in an upward shifted energyof 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 natureof 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 regionscontain nuclei where the ground state configuration including excitations across a shellgap is energetically favorable compared with the normal ground state configuration. Thisis an inverted situation because normally excitations across a shell gap causes a drasticincrease in energy, hence the name ’island of inversion’. Figure 1.5 shows one of theseregions for nuclei with atomic number Z = 10, 11, 12 and neutron number around N = 20.

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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 fromreference [7]

As seen in section 1.1.1, the location of the shell gaps can change for exotic nuclei, as isthe case for these neutron rich nuclei. Instead of exhibiting the normal magic numberN = 20, the neutron level vd3/2 is strongly shifted upwards towards the vf7/2 and vp3/2levels. This causes the shell gap to disappear between these levels at N = 20 and createsa gap at N = 16 between the vs1/2 and vd3/2 levels. This process can be seen on Figure1.6 for a 24O nucleus. In 30Si, 6 protons occupy the πd5/2 level. In 24O this proton levelis unfilled and the attractive tensor force between protons in the πd5/2 level and neutronsin 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 tensorinteraction. Picture taken from reference [8]

For the nuclei that lie in the island of inversion this effect is less pronounced due to thepresence of protons in the πd5/2 level, but it will still cause the magic number N = 20to weaken and will facilitate excitations across this shell gap. These excitations causethe presence of deformed 2p2h intruder states in the ground state wave function of thenucleus that are lower or comparable in energy to the predicted spherical 0p0h state bythe standard shell model theory [7].For nuclei with closed shells where there are no available orbits for the nucleons to move

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to, except for the ones across the shell gap, the ground state configuration is expected tobe spherical. This is because an excitation across the shell gap would normally requireso much energy that it would be near impossible to have a ground state configurationshowing such an excitation. But for the nuclei in the island of inversion, this is feasibleand the ground state exhibits a superposition of 2 configurations. One in which there isno nucleon excitation (0p0h) and one in which there are 2 neutrons that are excited fromthe sd-shell to the pf-shell [7]. This superposition of configurations is called configurationmixing. It originates from valence nucleons outside the closed shells that scatter eachother into other orbits [3].

Looking back at Figure 1.5, the nuclei in and around the island of inversion are labeledaccording to the configuration of their ground state. The red label pf means that for thesenuclei, 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 isimportant to keep investigating the surrounding nuclei. In the following sections it isshown how we can probe the ground state configuration and identify intruder states bylooking at the nuclear moments.

1.2 Nuclear electromagnetic moments

One way of looking at a nucleus is as a distribution of charges. This produces electricand magnetic fields inside the nucleus to which we can assign electromagnetic multipolemoments. For example a spherical point charge, like a single electron, would exhibit onlya monopole field. A more complex nucleus will have a more intricate structure, but willstill tend to symmetric shapes. This is why it is sufficient to consider only the lowestorder 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]

|µ| = e

2m|l| (1.5)

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 isreplaced by a magnetic moment operator µ. The observable moment µ is defined along thedirection of maximal angular momentum. Thus |l| has to be replaced by ml~ = l~, whichis the expectation value of the angular momentum operator along maximum projectiondirection. The quantum mechanical counterpart of equation 1.5 is given by the followingequation. [9]

µ = µz =e~2m

l (1.6)

Besides an orbital angular momentum l, nucleons also posses an intrinsic spin s. This spinalso creates a magnetic field and will also give a contribution to the magnetic moment.The following equations represent the two contributions [9].

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µ = gllµN µ = gssµN

gπl = 1 gπs = 5.5856912(22) (1.7)

gvl = 0 gvs = −3.8260837(18)

gl and gs are called the g-factors. The values above are determined for free protons andneutrons. µN is called the nuclear magneton and equals the quantity e~

2mwhen e and m

are the proton charge and proton mass.

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

µ(I) =⟨I(j1, j2, . . . , jn),m = I

∣∣∣ n∑i=1

µz(i)∣∣∣I(j1, j2, . . . , jn),m = I

⟩(1.8)

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

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

µ =

[(j − 1

2)gl + 1

2gs)]µN if j = l + 1

2

[j(j+ 3

2)

j+1gl − 1

2jj+1

gs]µN if j = l − 12

(1.9)

The Schmidt moments of equation 1.9 can be used to calculate the total magnetic dipolemoment for different possible configurations of a nucleus. However, it has been shownthat the presence of other valence nucleons changes the single particle moments due tomeson exchange interactions. The Schmidt moments have to be corrected for this effectbefore being used [10].

The magnetic moment of a nucleus is determined by the contributions of the valencenucleons, so its value is sensitive to the orbits that these nucleons occupy. The magneticmoment is thus a good probe for the purity of a configuration since the change of orbitoccupation 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 distributionfrom a spherical one. Its operator is defined as

10

Q = e(3z2 − r2) (1.10)

where e is the charge and z and r are the position coordinates of the orbit on whichthe nucleon moves [10]. Comparable to the magnetic dipole moment, the observablequadrupole moment is also defined as the evaluation of the operator along the directionin which the angular momentum has it maximal projection value. By convention thedirection is chosen as the z-axis.

For a doubly closed nucleus where the protons and neutrons are all paired, we expect tofind a spherical geometry. This means that the paired nucleons all move in sphericallysymmetric 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 thevalence nucleons outside the doubly closed shells [9]. Exceptions do exist however: someheavy mass nuclei exhibit deformed equilibrium shapes due to collective effects of thenucleons inside the closed shells. These nuclei have an intrinsic quadrupole momentdifferent from zero [2].

The contribution of a single nucleon located in orbit j is given by equation 1.11 of singleparticle quadrupole moments

Qs.p. = −ej2j − 1

2(j + 1)< r2j > (1.11)

where < r2j > is the mean square radius of the orbit j [2]. When there are multiple valencenucleons occupying orbit j, the quadrupole contribution from that orbit is determined by

Qj(n) = Qs.p.

(1− 2

n− 1

2j − 1

)(1.12)

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 themagnetic dipole moment: applying the moment operator Q given by (1.10) to each in-dividual nucleon and summing the result. This is again possible since Q is a one-bodyoperator:

Qs(I) =⟨I(j1, j2, . . . , jn),m = I

∣∣∣ n∑i=1

Qz(i)∣∣∣I(j1, j2, . . . , jn),m = I

⟩(1.13)

where n equals the number of valence nucleons [10].Because equation 1.13 only considers valence nucleons, the free-nucleon charges eπ = 1and ev = 0 have to be changed to effective charges eeffπ and eeffv . These effective chargeswill take into account the nucleon-nucleon and nucleon-core residual interactions thathave been left out in equation 1.13. So neutrons, that are originally expected to have nocontribution to the quadrupole moment because of their zero charge, now do contributedue to their non-zero effective charge.

11

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

Q0 =3√5πR2avZβ(1 + 0.16β) (1.14)

where Z is the atomic number, Rav is the average nuclear charge radius and β is adeformation parameter related to the eccentricity of an ellipse. If β is zero, Q0 is zero aswell and the nucleus has a spherical shape. [2]The intrinsic quadrupole moment can be determined by measuring the observable orspectroscopic quadrupole moment (Qs(I)). The link between these moments is given by:

Qs =3K2 − I(I + 1)

(I + 1)(2I + 3)Q0 (1.15)

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 ofthe intrinsic quadrupole moment. The deformation can be caused for example by thepresence of configuration mixing in which one of the configurations includes particle-holeexcitations across a normally closed shell gap (as is the case with nuclei belonging to theisland of inversion). By measuring the spectroscopic quadrupole moment, the amount ofdeformation and subsequently the amount of intruder state mixing in the ground state ofnuclei can be determined.

1.3 Large Scale Shell Model calculations

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

1.3.1 USD interaction

The first step in interpreting shell model calculations is defining the model space. Incase of the USD model, the shells up to the magic number Z,N = 8 are filled and areconsidered an inert core. This means that the nucleons in these orbits are assumed to befixed and cannot excite to other levels. The valence space is the sd-shell above this inertcore 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 isbounded by two shell gaps.

The nuclei that can be described by the USD model correspond to nuclei with atomicnumber 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 occupyorbits in the next pf-shell separated by a large shell gap. We’ve already seen that thereare nuclei in the island of inversion that have configurations in which this is indeed thecase 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 thiswill not be discussed in detail. For more information on the SDPF-interaction, see thework of Retamosa, Caurier, et al. in reference [12].

The next step is to solve the eigenvalue problem Hψ = Eψ with the nuclear Hamiltonianfrom 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 possibleconfigurations φ of the valence nucleons. The configurations are represented by Slaterdeterminants that are products of single particle states occupying different orbits. Thismatrix is then diagonalized to solve the eigenvalue problem [3]. Because the nuclearHamiltonian is rotationally invariant, the matrix elements conserve quantum number Jand M , corresponding to total angular momentum and its z-component. The matrixelements are thus determined by the possible orbit occupations of the valence nucleonsor SPE’s and the TBME’s between the nucleons that also preserve total J and M . Theenergy eigenvalues equal:

E(J) = ε1 + ε2 + VJ (1.16)

where εi are the SPE’s of the orbits the two nucleons occupy and VJ correspond to theTBME’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 residualnucleon-nucleon interaction part of the nuclear Hamiltonian.The valence nucleons can occupy any of the 3 orbits of the sd-shell. There are thus manymatrix 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 tothe 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 theexperimentally determined ones. There will most likely be a discrepancy due to theunknown exact form of the residual interaction V . The interaction is calculated frombasic meson exchange theory using the G-matrix formalism of Kuo [13]. Due to thediscrepancy, a χ2 analysis is performed and the TBME’s are adjusted. This process isrepeated until χ2 is minimized or reaches a certain minimum threshold [3].

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

Large scale shell model calculations are an essential tool for exploring the exotic regionsof the nuclear chart due to its predictive power. However, in this exotic frontier, magicnumbers 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 becomingcomputationally challenging for computer programs.To overcome this problem, a new method called Monte Carlo Shell Model (MCSM) hasbeen developed [14]. This method reduces the dimension of the H-matrix by selecting onlya 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 diagonalizationof the entire matrix. With MCSM, the former computational limits can be crossed anddimensions 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 analyzedover 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 theywere performed at the ISOLDE facility during the summer months of the year 2016. Thefollowing sections will discuss the isotope production and subsequent measurement withthe COLLAPS set-up. Also the analysis of the raw data and extraction of the hyperfineparameters from spectra will be described.

2.1 ISOLDE - isotope production

ISOLDE stands for Isotope Separator On-Line DEvice. In this building radioactive ionbeams are produced that are used in a variety of experimental set-ups scattered aroundthe 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 SynchrotronBooster (PSB) located nearby that produces protons with energies of 1.4 GeV. Theseprotons bombard a target material and create an array of radioactive nuclei throughnuclear reactions like fission, spallation and fragmentation [15]. To have a large yield ofcreated nuclei, the target has to be chosen well as to ensure a fast release of nuclei. Thisis why the target material is heated between 700◦C and 1400◦C. Typical release timesachieved are around 30 seconds. To achieve smaller release times down to tenths of asecond, the target material must be heated to above 2000◦C[16]. The atomic identity ofthe target also determines which nuclei are created in this process. For the aluminiumisotopes 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 isotopesfrom this experiment traveled from creation to arrival at the COLLAPS set-up. Picturetaken and adjusted from reference [25]

The released nuclei then diffuse to the ion source. ISOLDE uses three different types ofsources: 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 forthe experiment amongst the nuclei that were released from the target. The name RILISstands for Resonance Ionization Laser Ion Source and refers to the method by which itselects the isotopes. The ion source is nothing more than a hot cavity through whichmultiple 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 electronicexcitations of a specific element until this element is ionized. This method uses the uniqueatomic energy structure and has a large Z-selectivity [17]. The ionized nuclei are thenaccelerated 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 twomagnet systems in use. The GPS uses one bending magnet at an angle of 60◦. The HRSor High Resolution Separator uses two dipole magnets, first one at 90◦ and the secondat 60◦ angle. The analyzing magnet collects isotopes of certain mass at the focal planeof the magnet, while the isotopes of other masses are lost. The underlying principle isthe change in orbiting radius of an ion feeling a magnetic field depending on the massof the ion. During the measurements, different masses could be selected by increasing ordecreasing the magnetic field of the analyzing magnet. In this research project, the HRSis used with its superior mass resolving power M/∆M between 7000 and 15000 comparedto the GPS system with resolving power of only 2400. [18]

The isotopes collected at the focal plane of the magnet system are subsequently cooledand possibly bunched by a gas-filled linear Paul trap called ISCOOL. Figure 2.10 showsa 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 andisotope yield. The yields are all determined for UCx target and RILIS ion source. Datataken from ISOLDE yield database [20].

isotope half life yield (ions/C)26Al 7.2E5 y not listed26mAl 6.3 s 6.8E427Al stable not listed28Al 2.2 min 4.0E729Al 6.6 min 4.5E730Al 3.6 s 2.5E631Al 644 ms 2.5E5

pairwise coupled to an applied radio-frequency voltage that confine the ions in the radialdirection. Surrounding these rods are segmented DC electrodes that create a potentialgradient guiding the ions to the end of the ISCOOL ’tube’. The structure is also filledwith Helium gas at a pressure of 0.1 mbar that slows down the ions and reduces theirenergy 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 trappingpotential. In this way, bunches of ions can be released in succession instead of a continuousstream of ion release. By changing the accumulation time, the amount of ions in onebunch and the time window between releases can be determined by preference. In thisexperiment, the bunched beam mode was used for reasons that are linked to the operationof 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 andthe ion source, assuming that the energy of the incoming protons is fixed. There will beoccasional 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 thisexperiment the target is UCx and the ion source is RILIS. The yield of the aluminiumisotopes 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 bymatching 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. Thisis because the hyperfine interaction between the atomic electrons and the nucleus changesthe atomic structure. This effect is studied. To detect this change, a high resolution isrequired. Collinear Laser Spectroscopy or COLLAPS fulfills this requirement. In thissection 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 orbitssurrounding the nucleus. An atomic transition happens when an electron moves from oneorbit 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 definedby an electronic Hamiltonian:

He = Te + Uee (2.18)

where Te represents the kinetic energy and Uee the e− - e− Coulomb interaction, togetherwith a monopole interaction between the total charge of the nucleus and the electronsgiven by:

Q0 · V 0 = Une =−e2NZ4πε0re

(2.19)

Here −eN is the charge of the electron cloud, eZ the charge of the nucleus and re thedistance between nucleus and electrons. Q0 · V 0 represents the monopole part of the inter-action between the electrostatic multipole moment Q of the nucleus and the electrostaticmultipole field of the electron cloud generated at the nucleus V [21]. The electronic levelsobtained 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 itsinternal 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 Coulombelectrostatic potential between nucleus and electrons (charge-charge interaction). Theother is the movement and the spin of the nucleons that create respectively convectionand spin currents inside the nucleus. These nuclear currents can interact with currentsgenerated by the electron cloud (current-current interaction).

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

Hjj =∞∑n=0

B(n) ·M (n)

2n+ 1(2.20)

18

j in Hjj refers to the current density distribution, M (n) is the magnetic multipole momentof the nucleus of order n and B(n) is the magnetic multipole field at the nucleus generatedby the electron cloud. Due to the parity requirement of the nuclear state |I >, onlyodd 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 magneticdipole moment of the nucleus µI , B

(1) defined as the magnetic hyperfine field B(0) andthe current-current interaction rewritten as:

Hjj = −µI · B(0) (2.21)

Considering an atomic system including hyperfine interactions, the relative orientation ofnuclear angular momentum I and electronic angular momentum J has an effect on theenergy state of the atom. The atom can be described by a new quantum number F whichis the vector sum of I and J and takes on values from a parallel F = I+J to anti-parallelF = |I − J | orientation. Computing the matrix elements of the Hamiltonian of equation2.21 on an atom in a state |F > results in the energy contribution of the current-currentinteraction:

< F |Hjj|F >= Ejj =1

2AJC (2.22)

where C is a shorthand notation for F (F +1)−I(I+1)−J(J+1) and AJ is the magnetichyperfine coupling constant:

AJ =µB(0)

IJ(2.23)

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

Hqq =∞∑n=0

Q(n) · V (n)

2n+ 1(2.24)

Due to the parity requirement of the nucleus, only even n terms survive. This correspondsto contributions of the monopole term (n = 0) and quadrupole term (n = 2), assuminghigher order terms are negligible. Evaluating the Hamiltonian of equation 2.24 for nuclearstates |F > gives the charge-charge interaction energy contribution:

< F |Hqq|F >= Eqq = BJ3C(C + 1)− 4I(I + 1)J(J + 1)

8I(2I − 1)J(2J − 1)(2.25)

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 gradientat the nucleus generated by the electron cloud. From the definition of the quadrupolemoment given in section 1.2.2, it is known that Q is zero for nuclei with a spherical chargedistribution (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 gradientVzz will vanish. Thus, the charge-charge interaction energy given in equation 2.25 willonly 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 areadded 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 thefine 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 differencebetween fine and hyperfine structure with dummy values for I and J . The fine structurelevels undergo splitting as a result of the different orientations of I and J and the amountof splitting is determined by equations 2.22 and 2.25 together. The energy scale of thehyperfine splitting lies in the µeV range, while the fine structure splitting lies in the meVrange [21].

2.2.2 COLLAPS

After the bunching and cooling by ISCOOL, the ion beam is accelerated to an energy oftens of keV before being sent to the COLLAPS set-up. The availability of such an ionbeam provided by the ISOLDE facility is well suited for high-resolution laser spectroscopysuch as the collinear laser spectroscopy method. In this method, the ion beam is collinearlyoverlapped with a co-propagating laser beam. Photons from the laser are absorbed by theions if their energy matches an atomic transition. This resonance can be detected by thefluorescence photons emitted by the ions after decaying back to a lower-lying level. Bysweeping the laser frequency across a certain atomic transition, the hyperfine structure ofthe 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 between10 and 1000 MHz. To be able to distinguish between the obtained spectral lines of sucha structure, a very high resolution is needed [22]. The key to obtain this resolution, isthe acceleration of the ion beam. The line width of the spectral lines can be calculatedfrom the velocity distribution of the ions. Before arriving at COLLAPS, the ions havean energy spread δE of around 1 eV, thanks to ISCOOL. The velocity distribution of theions can be determined from the kinetic energy spread by the following equation [23].

δE = δ(1

2mv2

)= mvδv (2.27)

An acceleration voltage V gives an energy of eV to the ions. This acceleration will begiven to all ions equally, so the energy spread δE remains the same. Equation 2.27 showsthat the velocity distribution δv decreases if the velocity increases. The acceleration isthus the determining factor in decreasing the velocity distribution and also the line widthas 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 theDoppler-shift is given by ∆f = f0(v/c) where f0 equals the laser frequency. Due to thevelocity distribution, the ions will absorb the photons at different frequencies due to thedifferent Doppler-shifts for each velocity v. This frequency range determines the linewidth of the detected fluorescence photons and is called Doppler broadening. The linewidth or Doppler width is given by δf = f0(δv/c). Combining this equation with equation2.27 and using the relation v =

√2eV/m, the following expression for Doppler width is

obtained:

δf = f0δE√

2eV mc2(2.28)

where m is the mass of the ions. Realistic values inserted in formula 2.28 results in aDoppler 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 inchapter 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 frequencyof the fluorescence photon will have an inherent uncertainty determined by Heisenberguncertainty principle (δEδt > ~):

δfnat =δE

h=

1

2πτ(2.29)

with τ the lifetime of the excited state.Another contribution to the line width comes from power broadening. Opposite to thenatural line width that is determined by spontaneous emission, power broadening comesfrom stimulated emission. When the laser power is increased, more ions will be excited andmore fluorescence photons will be emitted. This increase in stimulated emission rate lowersthe lifetime of the excited level compared to the natural situation with only spontaneousemission. 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 laserpower P and saturation power Psat [24]:

δfpow = δf

√1 +

P

Psat(2.30)

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 thelaser frequency across the atomic transition frequency. A more accurate method is to fixand stabilize the laser to the optical transition frequency and use the Doppler-shift tochange 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:

f = f01∓ β√1− β2

(2.31)

with β being the relativistic Lorentz factor. The minus sign corresponds with co-propagatingbeams while the plus sign corresponds with contra-propagating beams. β equals:

β =v

c=

√1− m2c4

(eV +mc2)2(2.32)

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 andthus the scanning frequency [23]. This Doppler-tuning is the standard operating mode ofthe COLLAPS set-up.

22

Set-up

Figure 2.13: Schematic of the components of the COLLAPS experimental set-up includingthe 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 laserbeam 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 thatcan supply a voltage between −10 and +10 V used to scan the hyperfine structure. Thisvoltage is amplified by a linear-voltage amplifier called Kepco. Another voltage is suppliedby a high-voltage power supply called a Fluke. The Fluke voltage ranges between −10and +10 kV and is used to match the atomic transition frequency with the laser frequencyfor the different masses of the measured isotopes [25].

After acceleration the ions are brought into the Charge Exchange Cell (CEC). The CEC isa heated, vapour filled chamber where the singly charged aluminium ions are neutralizedtrough collisions with the gas atoms. In these collisions, the atomic electron clouds of the2 particles come into contact and one of the electrons from the gas atom is transferredto the ion, effectively neutralizing it. This neutralization is performed in cases where theionic 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 thisexperiment.

Al+ + Na→ Al + Na+ + ∆E (2.33)

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

23

Besides the elastic collisions, there will also be a certain amount of inelastic collisionswhere the aluminium particles lose some of their energy. This side-effect causes theemergence of sidepeaks in the spectrum that are shifted with respect to the main resonancepeaks. 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 of4 photomultiplier tubes or PMT’s. A photon that impacts the photocathode plate of thePMT, releases an electron through the photoelectric effect. This electron is multipliedinside the tube and the amplified signal is measured and stored. The main disadvantage ofthe PMT system used in the COLLAPS set-up is the low efficiency and especially the highnumber of background counts. This becomes troublesome when measuring exotic isotopesthat have a low production yield. There are several sources for background counts, thefollowing 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 ofresidual air left in the COLLAPS apparatus. Since the entire set-up is kept at highvacuum (10−7 mbar), these photons have an insignificant contribution. Photonscreated by interaction with the gas atoms from the CEC will have a much largereffect on the background. Light of this kind, produced in an area other than thelight collection region, can be suppressed by an aperture system. The laser and ionbeam travel superimposed through the apparatus with a certain circumference. Byplacing an aperture of certain size between the CEC and light collection region thatallows only the ion beam to pass, much of the stray light can be stopped beforereaching the PMT detection system. The PMT system including the aperture isshown on Figure 2.14. The picture also depicts a cooled housing, which was removedin this experiment [25].

Figure 2.14: Schematic of the fluorescence detection system showing the 4 PMT’s, theaperture set and lens system. The cooled housing shown in this picture was removed inthis 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 lightcan be produced inside the light collection region and thus cannot be stopped by theaperture. A way to reduce background counts in general is by operating ISCOOLin bunched mode. The PMT signal can be gated as to only allow photon countingwhen a bunch has reached the light collection region. In this way the temporalconstant background can be greatly reduced by measuring in certain time intervalswhen fluorescence photons are expected. The accumulation time in ISCOOL isset to create bunched ions with a temporal width of ±20 µs and temporal spacingbetween bunches of ±100 ms. Background photons can thus only contribute for aperiod of a couple of microseconds. The background can be suppressed by a factorof 103 − 104 (100ms/20µs) [26].

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

• At last, there are the dark counts that increase the background. Dark counts arecounts that the PMT detects when there are no photons hitting the phototcathodeplate. 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 ofthe PMT system (see Figure 2.14). The lenses focus the emitted fluorescence photons toimpinge on the photocathode plate. This increases the probability of a photon reachingthe plate. The location of the focal point between the lens and the PMT depends on thewavelength of the fluorescence photon. For practical reasons, the same lens should beused for every experiment [25].

2.3 Data

2.3.1 Files

The spectra of fluorescence counts collected by the COLLAPS experiment are gatheredand saved by the Measurement Control Program or MCP. The MCP-files together withthe lab book contain the raw data and all the information of the experiment necessary toperform an analysis. Table 2.2 shows an example for the measurement ’run234’ of 27Alof the information that can be extracted from the MCP-files. The photocounts are savedfor each PMT, gated and ungated. The photocounts used in the analysis is the sum ofthe gated photocounts collected by the 4 PMT’s.The spectra for which there were unstable parameters, e.g. drift in laser frequency orfluke voltage, are left out of the analysis. Sometimes a sequence of measurements areperformed 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 filesthat were used for the final analysis discussed in chapter 3.

25

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

Parameters from MCP file values for run 234 of 27Al

Line Voltage (V) (min. − max.) -2.5− 0.5Photocounts (#) (min. − max.) 34− 1414

Fluke (#) 1Fluke Voltage (V) 668.99698

ISCOOL Voltage (V) 29976.41806Timestamp 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 1727Al 97 7028Al 8 729Al 10 1030Al 10 431Al 15 3

2.3.2 P3/2 → S1/2 Transition

The atomic transition of aluminium measured in this experiment is the transition fromthe 3P3/2 to the 4S1/2 level. Figure 2.15 shows a simplified level scheme of aluminium andthe 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. TheP3/2 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 Einsteincoefficient (A = 1 · 108) than the P1/2 - S1/2 transition (A = 5 · 107) and is therefore a

26

stronger transition. The transition to the 2D doublet is also not preferred due to theproximity 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 lightof 792 nm is frequency doubled by the process of second harmonic generation to producelight of 396 nm used to excite the aluminium isotopes.

Figure 2.16 shows the transition between the two atomic levels including the hyperfinesplitting 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 thereference isotope 27Al with nuclear spin I = 5/2. The transitions between the hyperfinelevels are matched with the observed fluorescence peaks in a collected spectrum. Theallowed transitions between the hyperfine F levels are determined by the selection rulesof 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 spectrumshown underneath.

3 Analysis

The voltage in the MCP files is first converted to the observed frequency. Afterward theanalysis is performed using a custom written Python package called SATLAS (’StatisticalAnalysis Toolbox for LAser Spectroscopy’). It was created by PhD students Wouter Ginsand 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 forthe 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 ISCOOLcan be viewed as a downward slope along which the ions slide off and gain energy as canbe seen on Fig. 2.10. In contrast to this, the fluke and scanning voltage can be viewed asbarriers. 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. Themeasurements continued before the voltage had settled to its original value. Excludingthese values, the voltage is stable throughout the experiment, lasting several days, witha 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 atomictransition frequency with the laser frequency (see section 2.2.2). This is done by a voltagesupplied by the fluke. There were 3 flukes available for this experiment, but since fluke2 showed discrepancy between the read-out and the actual supplied voltage, only fluke1 and 3 were used for measurements. Figure 3.18 shows the stability of fluke 1 for thedifferent masses of the aluminium isotopes. Some figures show plateaus of stable voltages.These signify subsequent measurements that are interrupted by measurements of anothermass. When the original mass is measured again, the fluke voltage setting can be shiftedby a small amount. The large variation (3 V) for 26Al is due to a change in voltage settingon the fluke. The change in voltage is a result of the change in laser frequency lock thattook place between the measurement sequences of 26Al. It can be concluded that the flukeis stable during subsequent measurements and shows a variation of the order of 0.01 Vwhen this effect is neglected.

Figure 3.19 shows the stability of fluke 3. Also here plateaus of subsequent measurementsare 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 spectrumlies within the scanning voltage. This voltage is user dependent and can be found in theMCP-files. The scanning voltage is amplified by a kepcofactor k. Because the fluke is notgrounded, it supplies a voltage to the beam line with respect to the amplified scanningvoltage. Fluke devices are never 100% identical, so the amplification factor k is determinedfor each fluke separately. The kepcofactor k is determined by means of calibration curvesor ’kepcoscans’ that measure the amplified voltage k ·Vscan versus Vscan. Figure 3.20 showssuch a calibration curve.

Figure 3.20: Calibration curve of fluke 1.

There were 9 kepcoscans taken during the experiment. The average of the determinedslopes 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 convertedfrom voltage to frequency. This can be done by filling in the variables in equations 2.31and 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 thehyperfine 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 andextracting the hyperfine parameters A and B in units of MHz. For this, a basemodel hasto be created to represent the experimental data. Then the free parameters of the modelare adjusted in the fitting procedure such that the model best matches the data. Thisis the basic premise of the fitting routine and is discussed in more detail in the followingsections.

3.2.1 BaseModel

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

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

• 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 valuesgiven as input are either rough estimates or taken from literature if available. Duringthe 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 thespectrum with respect to the transition frequency. It can be estimated as the offsetof 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,jin frequency space can be determined based on the energy difference between 2 hyperfinelevels 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 functionafter fitting is based on the minimization of the cost function. The cost function representshow well the model function with its current set of parameters agrees with the data. Ahigh value indicates more disagreement than a low value. This is why the cost functionis minimized during fitting by altering the free parameters until the set of parameters isachieved for which the cost function has reached its lowest value. This set of parametersagrees best with the data and the fitting routine ends.The minimization of this function during fitting is done by altering the free parameters ofthe model until the best model and parameters are obtained that cannot agree more withthe data and the fitting routine comes to an end. The SATLAS package uses the quicklyconverging Levenberg-Marquardt algorithm for the minimization of the cost function inthe case of a sum-of-squares function. The cost function itself is the weighted χ2 formula:

χ2 =N∑i=1

(yi − f(xi)

σi

)2(3.39)

where yi is the experimental data, N is the amount of data points, f(xi) is the modelvalue at frequency value xi and σi is the uncertainty on each data point. Because this isa 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:

χ2red =

χ2

N −#(freeparameters)(3.40)

where N indicates the amount of data points. A value of 1 means perfect agreementbetween model and data. In this experiment, the χ2

red values were consistently slightlyhigher than 1 which signifies an underestimation of the errors and that the fit has notfully captured the data. But only when the values are greatly higher than 1 a poor modelfit is achieved.

33

Figure 3.22 shows an example of a fitting routine for 27Al, where the left figure shows themodel 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 ofa number of parameters for each measurement. The parameters include the hyperfineconstants A and B and the centroid, but also other parameters that are discussed lateron 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 takenfrom the independent measurements. This average takes into account the error on eachindependent measurement and is given by equation:

〈x〉weight =

∑Ni=1

xiσ2i∑N

i=11σ2i

(3.41)

where the weights are the reciprocal of the error variance σ2 of each measurement. Inthis way measurements with large errors have less effect on the final result than moreaccurate measurements with small errors. The standard deviation σ is provided for eachmeasurement by the fit program.

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

σstat =

√1∑Ni=1

1σ2i

(3.42)

and a scattering error [32]:

σscatt =

√√√√√∑Ni=1

(xi−〈x〉weight)2

σ2i

(N − 1)∑N

i=11σ2i

(3.43)

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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 maximalof 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 dependenton the initial quality of the basemodel. Besides the input parameters that are used tocalculate the position of the transitions, there are also other optional parameters thatprimarily influence the shape of the basemodel spectrum used for fitting. The optimalsettings of these parameters were determined by variation until the best fitting resultswere obtained with respect to the data in this experiment. Some of these parameterscould be estimated from theoretical considerations, others had to be determined by trialand 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 previoustwo. To determine these parameters, an analysis of the linewidth can be performed. Insection 2.2.2, the different contributions to the spectral linewidth of the atomic transitionwere discussed. There are 3 major contributions with their own shape and width:

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

• Natural linewidth: Equation 2.29 is evaluated by determining the lifetime τ ofthe excited state. This can be done by considering the Einstein coefficient A forspontaneous emission. For the transition 4S1/2 - 3P3/2, this value is 9.85 · 107 s−1.Since A = 1/τ , the lifetime is approximately 1 · 10−8 s [30]. The natural linewidthbecomes about 16 MHz and has a Lorentzian shape.

• Power broadening: Equation 2.30 can be evaluated by finding the saturation laserpower at which point increasing the power does not result in more photon emissionbecause 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 notvery clear due to the low amount of data points, but it can be deduced that thesaturation power is around 2.1 mW. This estimation is enough since the goal is tohave an estimated input FWHM value for the basemodel. The laser power used inthe measurements was 1.9 mW, so the effect of power broadening is an increase of38% compared to the original linewidth. This means that the Lorentzian naturallinewidth 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 contributiondue 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 andLorentzian contribution respectively. The width of the Voigt profile is given by [31]:

fV = 0.5346fL +√

0.2166f 2L + f 2

G (3.44)

where fG and fL are the Gaussian and Lorentzian linewidths. This gives an estimatedVoigt width fV of 26 MHz.

After the first analysis of the stable isotope 27Al, an average FWHM value of 173 MHzwas obtained, which is by far larger than the theoretical expected value. One explanationis the presence of sidepeaks in the spectrum. Sidepeaks originate from inelastic scatteringevents between the ions and gas atoms in the CEC. It causes the ions to populate adifferent excited state and create spectrum peaks at some frequency offset left of themain peaks. This asymmetric broadening thus increases the linewidth. When correctedfor 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, thepeaks of the hyperfine spectrum are all resolved and the the hyperfine constants A andB can be determined.

The analysis of 27Al shows that the Voigt lineshape leads to the best fit of the data. Theaverage χ2

red value was 2.44 for the Gaussian shape, 1.61 for the Lorentzian shape and1.42 for the Voigt shape.

36

Sidepeak parameters

As mentioned above, sidepeaks can appear in the hyperfine spectrum and have an effecton the quality of the fit and the obtained fit parameters. It originates from followinginelastic 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 fastAl-ions or Al-atoms that transfer momentum onto the Na-atoms and lose some of theirvelocity. This momentum transfer is indicated by the loss of energy ∆E. These inelasticcollisions can happen to the same Al-atom several times. The probability of the sameatom undergoing N successive collisions is given by Poisson’s law since the collisions areidentical and independent:

P (N) =xN

N !e−x (3.47)

where x is a parameter dependent on the length and density of the Na-vapor [33]. Forhigh 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 asN satellite or ’side’ peaks at the low energy side of the main peak with an offset equalto N ·∆E. The SATLAS package allows to choose the amount of side peaks N which isnot varied during fitting. It also allows to give an initial value to the relative intensity ofthe first side peak while the other side peak intensities are calculated using the Poissondistribution (see equation 3.47). The offset frequency of the first side peak can also begiven and is varied during fitting.

Through a fit trial, the energy defect ∆E and thus amount of offset and the amount ofsignificant side peaks can be determined. Figure 3.24 shows the evolution of the χ2

red

when the fitting is performed with increasing amount of sidepeaks. It shows that asignificant improvement is achieved up to an amount of 4 sidepeaks. This means thereare 4 sidepeaks that influence the spectrum. In reality there are even more side peaksbut due to the Poisson probability, their intensity drops sharply and do not significantlyaffect the spectrum.

37

Figure 3.24: Average χ2red values resulting from SATLAS fit procedure for different

amounts of side peaks. A horizontal line is drawn through the value of 4 side peaksto guide the eye.

To determine the offset, the variable offset can be compared between fits of different sidepeak amounts. Figure 3.25 shows the offset for each measurement for varying numberof sidepeaks fitted. With increasing amount of side peaks fitted, the offset is more andmore uniform between measurements. Since the offset is not a random value, it shouldbe the same for different measurements. This means that the offset value given by thefit procedure is only reliable starting from 4 sidepeaks where the scatter in offset hasdecreased. Table 3.4 shows the offset values calculated as a weighted average and theerror. 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 measurementsof 27Al. The offset is compared between fit procedures with varying amount of sidepeaksfitted.

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 offsetof 3.6 eV. This suggests that the inelastic scattering involves excitation to to 4P tripletlocated at this energy [34]. The energy offset is independent of mass and can be used tocalculate the frequency offset for each isotopic mass under investigation using the formulasfor frequency conversion: equations 2.31 and 2.32. Now the offset can be fixed for eachmass during the fit process.

The importance of the sidepeak correction for the fit results can be seen in the values ofthe hyperfine constants. Figure 3.26 shows the result of the magnetic hyperfine constantAlower of the P3/2 atomic state for a different amount of sidepeaks considered duringfitting. 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 ofsidepeaks included in fit. Literature value taken from [35].

Other parameters

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

The intensities of the peaks were given an equal initial value and were varied duringfitting. There is the option to account for saturation in the spectra. Since none of theexperimental spectra showed saturation, there was no difference in the fitting result whensaturation 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 obtainedfit variables, some parameters can be fixed to a known value. One of those parameters isthe ratio of the hyperfine constant Au of the S1/2 level to the hyperfine constant Al of theP3/2 level.Along an isotopic chain, this ratio has a constant value. This can be seen from equation2.23: the values of µ and I are the same for Au and Al, so the ratio is determined bythe values of the hyperfine magnetic field B(0) and the electronic spin J of the P3/2 andS1/2 atomic state. Between different isotopes the ratio of J remains the same, since thesame transition is investigated in each isotope. Also the value of B(0) is the same becauseonly the neutron number changes between isotopes: the electron cloud that creates themagnetic 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 isadded between brackets.

Isotope (I) A-ratio

26Al (5) 4.486 (14)27Al (5

2) 4.572 (5)

28Al (3) 4.567 (5)29Al (5

2) 4.573 (2)

30Al (3) 4.562 (12)31Al (5

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 thisisotopes does not agree with the average A-ratio. With fixed ratio, there was a reductionof 0.06 in the error of Al. This reduction is not observed for Au. On the other hand theBl value moves further away from the literature value and does not correspond with itanymore within a 1σ error.It was decided to not constrain the A-ratio since the improvement in error on Al does notweigh 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 hyperfineanomaly is negligibly small. The hyperfine anomaly takes into account the distribution ofthe 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 interactionthat manifests itself in the value of the hyperfine constant A. This hyperfine anomaly isdifferent for different isotopes, which causes the A-ratio to not remain constant along theisotopic 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 thehyperfine anomaly is indeed negligibly small.

4 Results

This chapter contains the results regarding the hyperfine constants obtained by the fittingprocedure described in the previous chapter. The magnetic dipole moment µ and electricquadrupole moment Q can be calculated through the literature values of the referenceisotope 27Al and the data on the hyperfine constants obtained in this experiment. Theaccuracy of the determination of nuclear moments with the COLLAPS method is inferredfrom the comparison with methods used in literature. Finally, the nuclear moments of29Al 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 asa result of the fit process. The values and errors are calculated using equations 3.41 and3.43. Literature values are found only for 27Al and 26Al.

Table 4.6: Summary of hyperfine constants A and B of the 3s23p 2P3/2 level and the 3s24s2S1/2 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 valuesexcept for the Bl value of 27Al. The experimental result underestimates this value. Thehyperfine constant Au of the S1/2 state of 27Al determined in this experiment showsimprovement in accuracy over the literature value with a large decrease in uncertaintyfrom 15 to 0.1 MHz. Also the experimental errors on Al and Bl of 26Al are smaller but ofthe 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 becalculated. Additionally, the nuclear moments of one of the isotopes is also required toserve as a reference. In this experiment the nuclear moments of the stable 27Al isotopeis used from literature and we can consider the ratio between the magnetic moment of acertain isotope and the reference isotope:

µ

µref=

IJAB(0)

IrefJrefAref

B(0)ref

(4.48)

The ratio resulted from using equation 2.23. The electronic spins J and Jref cancel ifboth magnetic moments are calculated using the hyperfine constant A of the same atomicstate: P3/2 or S1/2. The magnetic fields B(0) and B(0)ref are equal and also cancelsince the electron cloud is the same for each isotope. Now an equation is achieved todetermine µ that is only dependent on the hyperfine constants (experiment), the nuclearspins (literature) and the reference magnetic moment (literature):

µ = µrefAI

ArefIref(4.49)

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For the calculation of the magnetic moments with equation 4.49 always the Au hyperfineconstant of state S1/2 is used in this experiment. Its value is significantly higher than thatof Al of state P3/2 and since the uncertainties on both constants are of the same order ofmagnitude, 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:

Q

Qref

=BeVzzBref

erefVzz/ref

(4.50)

The charges e and eref are both the same and equal to the elementary charge. The electricfield gradients Vzz and Vzz/ref are both generated by the electron cloud. Since the electroncloud is the same for different isotopes, these values cancel. After simplification, equation4.50 leads to the following equation:

Q = QrefB

Bref

(4.51)

For the calculation of the quadrupole moments, the Bl hyperfine constant of state P3/2 isused 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 thealuminium 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 literatureis confirmed for each aluminium isotope, except for 26Al and 31Al. The experimentalvalues of 26Al and 31Al fall outside the errorbars of the literature value with a margin ofrespectively 0.011 and 0.001. The disagreement of 31Al could be explained by the statisticsand quality of the measurements, which is explained further in section 4.2.2. The valueof 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 andodd A isotopes. The odd A isotopes contain an uneven amount of protons (13) andalways an even amount of neutrons. The similarity of the magnetic moment for the oddisotopes illustrates the sensitivity of the magnetic moment to the orbital occupied by thelast unpaired nucleon. In this case the proton occupies the 1d5/2 orbital in the nuclearshell model. In fact it is the g-factors of the nuclei that show the similarity, but since thenuclear spin of 27Al, 29Al and 31Al are all equal to 5/2, the magnetic moments are thesame as well. The g-factor is obtained by simply dividing the magnetic moment by thenuclear spin I.

Figure 4.28 compares the experimental g-factor values for the odd A isotopes with theg-factor for an unpaired nucleon on the d5/2 orbit calculated using the Schmidt momentsgiven by equation 1.9. The first calculation was done using the free g-factors of equation1.7. The second calculation was done using the effective g-factors. These g-factors takeinto account the presence of a nuclear medium and correct for meson exchange interactionsand core polarization effects [10]. In this region of the nuclear chart, the values used fora proton are: gleff = 1.1 · glfree, gseff = 0.7 · gsfree [39]. As can be seen on Figure4.28, 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 presenceof configuration mixing and the fact that the single particle picture is in this case toosimplistic to fully predict the correct g-factor.

44

Figure 4.28: Experimental g-factors of the odd A isotopes in sequence 27-29-31. Errorbars are too small to be visible on the figure. Also plotted are the Schmidt g-factors for anucleus with unpaired nucleon in d5/2 orbital. The red line represents the Schmidt valuecalculated 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-factoris influenced by interaction between the 2 nucleons and the orbits they occupy and is thusnot 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 experimentalvalues correspond with literature within errorbars, except for 28Al and 31Al. For 29Al and30Al 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 onthe magnetic dipole moment. An explanation for this can be found in the magnitude ofthe hyperfine splitting caused by the dipole and quadrupole interaction. The quadrupoleinteraction has a smaller effect on the hyperfine structure of aluminium. Figure 2.16shows that the coefficients of the B constants are consistently smaller than those of theA constants. This makes the determination of B less accurate. The larger absoluteuncertainty on B together with the fact that B is smaller in value than A translates in alarge relative error. This relative error on B causes the larger error on Q compared withµ.

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 forthis deviation. 31Al has only 3 independent measurements to use in the analysis. Alsothese 3 spectra do show some amount of scatter in the peaks as can be seen on Figure4.30. Due to the low available amount of spectra, there was no luxury to dismiss thesemeasurements. This could be the reason of the deviation in quadrupole moment andmagnetic 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 comparingthe COLLAPS method with other methods used to determine nuclear moments in theliterature. The literature values for the nuclear moments listed in Table 4.7 and Table4.8 were determined using different methods:

The moments of 26Al were measured by atomic beam laser spectroscopy (ABLS), similarto 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 fromanother 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 sameorder of magnitude as in literature. The value on the other hand deviates from literatureby a small amount as seen in section 4.2.1. This deviation could be explained by thetransition mixing due to the proximity of the D5/2 to the D3/2 level, which would makethe literature value less precise. Another explanation could lie in the higher FWHM ofthe peaks obtained in this experiment compared with the literature experiment (110 MHzvs 75 MHz). A high FWHM results in a less precise determination of the position of thepeaks in the spectrum. This would indicate that the experimental value is further fromreality than the literature. Further investigation would have to be made to determinewhich value is closer to reality.The experimental quadrupole moment of 26Al agrees with literature and has a betteraccuracy. This could be due to the larger amount of measurements compared to theliterature experiment (17 vs 11).

The stable isotope 27Al has remarkable lower uncertainties than the other isotopes. Thetechniques used to determine the nuclear moments involve the investigation of aluminiumembedded in molecules and is described in articles by Kello V. et al. [42] and EpperleinB. 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 Larmorfrequency 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 inmolecules 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/quadrupoleresonance 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-decaypattern. The magnetic dipole moment is determined by matching the frequency of anapplied rf-field to the Larmor frequency. This will cause equalized population of thesubstates and destruction of the asymmetry which is expressed by a peak or dip in thespectrum. The accuracy of the technique is, like for COLLAPS, based on how well theposition 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 thedetermination of the quadrupole coupling frequency as well as the use of the Larmorfrequency 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 axiscan cause the resonance frequency that is found to deviate from the actual frequencywhich leads to a shift in the quadrupole moment determination. These effects cause thequadrupole moment extracted in this way to be less accurate than the magnetic momentdetermined through β-NMR.The experimental nuclear moments of 28Al and 30Al have comparable uncertainties tothose determined by the NMR-techniques of literature. The moments of 31Al are lessaccurately determined in this experiment. This might be caused by the low statisticswith only 3 measurements.

In conclusion, the results of this experiment by the COLLAPS set-up achieve similaraccuracies as the methods used in literature. Results from other experiments using theCOLLAPS set-up ([44]) to determine nuclear moments show uncertainties for the mag-netic dipole moment comparable to literature values obtained by other ABLS methodsand β-NMR. If we look at equation 2.23, we see that there is a general improvement inrelative error on the hyperfine constant A for heavier nuclei. Heavier nuclei have largermagnetic 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 smallerthan 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.

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4.4 New Results: 29Al and 30Al

The experiment has led to some new results, namely a value for the magnetic dipolemoment of 29Al and quadrupole moment for 29Al and 30Al. Although these values cannotbe compared to literature values, they can be compared to theory. Using large-scale shellmodel calculations, theoretical values for the nuclear moments can be found. Calculationsare performed using the USD interaction discussed in section 1.3.1 and the SDPF-Minteraction. The USD interaction valence space limits itself to the sd-shell, while SDPF-Malso 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 thepf-shell. Both interactions were calculated using the KSHELL program implementing afull diagonalisation, similar to the ANTOINE code.Comparison with these two interactions provides information about the ground stateconfiguration of our isotopes and allows us to determine their position with respect to theisland of inversion.

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

Figure 4.31: Comparison between experimental and calculated magnetic moments of27, 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 29Alcorresponds with the theoretically predicted value. The correspondence between USD andSDPF-M calculated values shows that the inclusion of the pf-shell makes no difference forthe magnetic moment. This means that there are no significant excitations by neutronsfrom the sd to pf orbitals in the ground state configuration.

Besides the magnetic moment, the quadrupole moment is also and even more sensitiveto the the presence of intruder states across the shell gap at N = 20. This is becauseof 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 forthe normal sd-shell ground state. Since the quadrupole moment is a measure of nucleardeformation, 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 theoreticalvalues which are most suited to use in sd-shell calculations [45].

Figure 4.32: Comparison between experimental and calculated quadrupole moments of27, 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 thepresence of intruder states and can be used to determine the amount of mixing betweensd and pf configurations in the ground state of the aluminium isotopes [45]. 27Al is alreadyconfirmed to be normal sd-configuration, while 31Al is mainly sd-shell configuration withpossible 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 theexperimental value. The uncertainty of the literature value is smaller and a more accuratecomparison with theory can be made than with the experimental value. The literaturevalue 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 froman article by Heylen, H. et al. ([46]). In this case, there is a clear difference between theQ 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 includesneutron excitations to the pf-shell. These neutrons can contribute to the quadrupolemoment 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 intruderstates 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 that29Al and 30Al both show no sign of intruder mixing and can be identified as purely normalsd-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 startsat 33Al.

Table 4.9 contains an overview of the experimental and calculated values used in Figures4.31 and 4.32.

Table 4.9: Overview of experimental and calculated nuclear moments for a sample ofaluminium isotopes. Experimental and calculated Quadrupole moments of 33Al are addedand taken from an article by Heylen, H. et al. ([46]). Magnetic moments are calculatedwith 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 bymeans of the COLLAPS beam line. In contrast to previous studies where the transitionfrom the P-doublet to the D-doublet was used ([41], [37]), the transition P3/2 - S1/2 wasused to create the hyperfine spectrum in this experiment. This eliminates transitionmixing between the D-doublets.

The achieved results for the nuclear moments mostly match with literature values withinerrorbars. This experiment also led to the determination of the previously unknownnuclear moments of 29Al and unknown quadrupole moment of 30Al.The magnetic hyperfine constant A of S1/2 for 27Al shows an improvement in accuracy by2 orders of magnitude compared to literature and should be used in future references. Alsothe quadrupole moment of 26Al has been more accurately determined with an uncertaintyhalf as small as determined in previous experiments.

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

6 References

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