Collectivity in Neutron-Rich Erbium Isotopes

(1)

Aila Gengelbach

Collectivity in Neutron-Rich Erbium Isotopes

Licentiate Thesis in Physics

(2)

Abstract

Neutron-rich rare-earth nuclei around the maximum of collectivity are predicted to exist with an extremely stable intrinsic configuration in their ground-state structure.

Due to the high degree of axial symmetry and large deformation, these nuclei are also excellent candidates for having long-lived high-K isomers. The present work concerns a study of the structure of the yrast bands and a search for isomers in the neutron-rich ₆₈Er isotopes. Excited states of ₆₈Er isotopes were populated via multi-nucleon transfer reactions.

A 859 MeV ¹³⁶Xe-beam was used to bombard a self-supporting ¹⁷⁰Er-target. The experimental setup consisted of the high-resolution γ-ray spectrometer AGATA coupled to the heavy-ion magnetic spectrometer PRISMA. The experiment collected 2 TB of useful data corresponding to 3 days of effective beam time.

Beam-like fragments were identified by the PRISMA spectrometer placed at the grazing angle of 44^◦. PRISMA allows for Z, A and q identification as well as TOF and velocity vector determination. This is required for the Doppler correction of the emitted γ rays detected in time coincidence with AGATA. A good Z and very clean A separation has been achieved in PRISMA. Making use of two-body kinematics, Doppler corrected γ-ray spectra for target-like fragments were obtained as well.

Due to the novel techniques of PSA and γ-ray tracking, AGATA provided high-quality γ-ray spectra for both beam-like xenon and target-like erbium isotopes. Known yrast bands and isomeric states in neutron-rich erbium isotopes were observed. A candidate for the decay of an isomeric state with Eγ= 184 keV in¹⁷³Er, which has no previously known excited states, was identified.

(3)

Svensk Sammanfattning

Kollektivitet i neutronrika erbiumisotoper

Redan i antiken funderade indiska och grekiska naturfilosofer över vilka byggstenar världen omkring dem är uppbyggd av. Naturens kretslopp av förfall och återskapande lade grunden till deras atomistiska världsbild:

konceptet att hela universum består av små odelbara enheter, så kallade atomer, i tomrummet.

En del av fysiken undersöker materiens minsta beståndsdelar och hur de växelverkar på ett mest grundläggande sätt. Dagens moderna atomte- ori uppstod under 1800-talet genom upptäckten av kemiska grundämnen och slumpmässiga rörelser, så kallade Brownsk rörelse, hos små damm- partiklar i vattendroppar. Dessa rörelser förklarades genom kollisioner mellan dammpartiklarna och små osynliga vätskepartiklar. Man förstod då att grundämnena i materian består av enskilda partiklar, atomer.

Atomen visade sig dock vara delbar och består av elektroner, som kret- sar i distinkta omloppsbanor kring atomkärnan. Denna modell kallas för Bohrs atommodell. Atomkärnan består i sin tur av två olika sorters nukleoner: protoner och neutroner. Antalet protoner bestämmer till vilket grundämne atomkärnan tillhör. Kärnor av samma grundämne men med olika antal neutroner kallas för isotoper. En nuklid är en atomkärna med ett specifikt antal neutroner och protoner. I atomkärnan hålls nuk- leonerna ihop av den starka växelverkan, som övervinner den repulsiva elektromagnetiska kraften mellan två lika laddade protoner.

Kärnstrukturfysik utforskar den starka kraften som verkar i atomkär- nan och mellan dess byggstenar. En av huvudfrågorna inom detta forskn- ingsområde är: Hur många eller hur få neutroner kan varje grundämne innehålla? Inom vilka gränser av antalet protoner och neutroner förmår den starka kraften att binda ihop nukliderna?. Teoretiska beräkningar tyder på att det finns mer än dubbelt så många än dagens kända ca 3500 nuklider. Med “känd” menas här att en nuklid har (producerats och) identifierats i laboratoriet. Ett mål med grundforskningen inom kärnstrukturfysik är att utforska de okända nukliderna för att kunna dra slutsatser om den starka växelverkan och atomkärnans inre struktur.

Partikelstruktur och kollektiv struktur är två modeller som beskriver olika aspekter av atomkärnans struktur. Atomkärnan lyder under kvant- fysikens lagar, vilket leder till att nukleoner organiserar sig i “slutna skal”, separata sådana för neutroner och protoner. Denna modell kallas

(4)

för skalmodell, där varje skal innehåller ett distinkt antal nukleoner. Vid helt fyllda (slutna) skal befinner sig atomkärnan i en särskilt stabil kon- figuration. Enstaka nukleoner inom slutna skal har lite inverkan på atom- kärnans struktur. Istället bestäms strukturen av nukleoner utanför dessa slutna skal.

Ju fler nukleoner som finns utanför de slutna skalen, desto fler möj- ligheter till växelverkan mellan dessa nukleoner finns. Beskrivningen genom partikelstrukturen bryter ihop. Istället uppstår regelbundna struk- turer som avspeglar kollektiva rotationer eller vibrationer. En jämn utveckling av dessa rotations- eller vibrationsfenomen över flera olika nuklider kännetecknar kollektiva kärnor. Enstaka nukleoner påverkar kärnstrukturen bara subtilt.

Denna avhandling undersöker kollektiva effekter i erbiumisotoper. Dessa isotoper ligger i närheten av¹⁷⁰Dy, en av de kanske mest kollektiva kärnor som ligger mitt emellan två slutna skal både för protoner och neutroner.

Kärnor i dess omgivning är neutronrika och måste skapas artificiellt i ett laboratoriet. Undersökningen i denna avhandling baseras på data från ett experiment i Italien. En jon-accelerator skapade en energirik

136Xe-stråle som skickades mot ett ¹⁷⁰Er-mål. På grund av den höga energin nuddar dessa strål- och målkärnor vid varandra och byter ut nukleoner. Detta kallas för flernukleon-överföringsreaktion. I utbytet överförs även rörelseenergi från strålen som leder till en omorganisation av atomkärnors struktur, vilket skapar exciterade tillstånd i både strål- och målliknande nuklider.

Denna avhandling studerar kollektiv kärnstruktur i erbiumisotoper och söker efter tidigare okända långlivade exciterade tillstånd, som även kallas “isomera tillstånd” i dessa isotoper. För att kunna analysera exper- imentet måste vi veta vilka nuklider och vilka exciterade tillstånd som har skapats. För detta har två olika instrument använts.

Det första instrumentet är den magnetiska spektrometern PRISMA, som mäter proton- och nukleonantalet för de strålliknande reaktionsproduk- terna samt storleken och riktningen av deras hastiget. Med hjälp av denna information beräknas de målliknande reaktionspartnernas proton- och nukleonantal, samt deras hastighet.

De exciterade tillstånd, som reaktionspartnern skapas i, sönderfaller till nuklidens grundtillstånd genom att skicka ut en kaskad av ljuspartik- lar, gammastrålning, med specifika energier. Dessa energier skiljer sig åt mellan olika nuklider och fungerar som ett “fingeravtryck” av nukliden och dess kärnstruktur.

Det andra instrumentet är gammaspektrometern AGATA, som bestäm- mer gammastrålningens energi och riktning med hög precision genom en ny spårningsteknik. När gammastrålningen träffar AGATA, så sprids den inom gammaspektrometern och lämnar en viss energi vid varje interaktionspunkt. Eftersom detektorerna i AGATA är segmenterade och position-

(5)

skänsliga, mäts energin och positionen för varje interaktionspunkt. En spårningsalgoritm återskapar därefter gammastrålningens spridningsväg i AGATA samt positionen för den första interaktionspunkten.

Dopplereffekter, vilka uppstår eftersom gammastrålningen skickas ut från nuklider i rörelse, gör att energin som registrerars i AGATA beror på vinkeln mellan gammastrålningens och de strål- och målliknande reak- tionsprodukternas riktning samt på deras hastighet. Gammastrålnin- gens riktning bestäms av positionen för den första interaktionspunkten i AGATA. Riktningen och hastigheten av de strål- och målliknande reak- tionsprodukterna bestäms med PRISMA. Därmed kan den uppmätta gam- maenergin i AGATA Dopplerkorrigeras. Utan Dopplerkorrigeringen skulle energiupplösningen försämras avsevärt. Med denna nya spårningsteknik har AGATA en mycket större känslighet än tidigare gammaspektrometer för detektering av svaga och sällsynta gammaövergångar i nukliderna.

Analysen som presenteras i denna avhandling bekräftar förekomsten av olika xenon- och erbiumisotoper genom deras unika “fingeravtryck”.

Utöver detta har en kandidat hittats för ett sönderfall av ett isomert till- stånd i den neutronrika¹⁷³Er-isotopen för vilken inga tidigare exciterade tillstånd är kända.

(6)

(7)

Oma, du fehlst

(8)

Acronyms

ADC analog-to-digital converter ADL AGATADetector Library

AGATA Advanced GAmma Tracking Array DAQ data aquisition system

FPD focal plane detector

FWHMfull width at half maximum GTS global trigger system

HPGehigh-purity germanium IC multi-anode ionisation chamber

INFN-LNLInstituto Nazionale di Fisica Nucleare - Laboratore Nazionali di Legnaro

MCP micro-channel plate

MWPPACmulti-wire parallel plate avalanche detector

Narvala software framework written in ADA for parallel processing on different computers

PRISMAheavy-ion magnetic spectrometer with large acceptance angle PSA pulse-shape analysis

TDC time-to-digital converter TOF time of flight

femul offline Narval emulator femul libPRISMA PRISMApreprocessing library

(9)

List of Figures

2.1 Multi-nucleon transfer . . . 22

3.1 Photograph of the experimental setup . . . 24

3.2 PRISMAscheme . . . 26

3.3 MCP scheme . . . 28

3.4 AGATA demonstrator . . . 32

3.5 Segmentation of an AGATA HPGe crystal . . . 33

3.6 DAQ scheme . . . 36

4.1 First interaction point in AGATA . . . 39

4.2 Un-/calibrated MCP . . . 41

4.3 Raw MWPPAC signals . . . 42

4.4 Calibrated xFP . . . 43

4.5 Calibrated TOF vs. xFP . . . 44

4.6 Raw IC and veto signals . . . 45

4.7 LISE++simulation . . . 46

4.8 Calibrated IC rows . . . 47

4.9 Z identification . . . 48

4.10 Ion trajectory reconstruction – dipole entrance . . . 52

4.11 Ion trajectory reconstruction – dipole . . . 54

4.12 Ion trajectory reconstruction – focal plane . . . 56

4.13 Charge states . . . 61

4.14 Mass to charge ratio for each charge state . . . 62

4.15 Calibrated mass spectrum . . . 63

4.16 ICtime alignment . . . 64

4.17 Time alignment of m/q . . . 65

4.18 MCP corrected m/q . . . 66

4.19 xFP corrected m/q . . . 67

4.20 Fully corrected mass . . . 68

4.21 Doppler correction . . . 70

4.22 Coincidence time spectrum . . . 73

4.23 Velocity distribution . . . 74

5.1 Normalised Eγ spectra . . . 78

5.2 Prompt ¹³⁶Xe γ-ray spectrum . . . 80

5.3 Prompt ¹⁷⁰Er γ-ray spectrum . . . 81

5.4 Delayed ¹⁷⁰Er γ-ray spectrum . . . 82

5.5 Prompt ¹³⁸Xe γ-ray spectrum . . . 83

5.6 Prompt ¹⁶⁸Er γ-ray spectrum . . . 83

5.7 Delayed ¹⁶⁸Er γ-ray spectrum . . . 84

(12)

5.8 Prompt ¹³⁷Xe γ-ray spectrum . . . 86

5.9 Prompt ¹⁶⁹Er γ-ray spectrum . . . 86

5.10 Delayed ¹⁶⁹Er γ-ray spectrum . . . 87

5.11 Prompt ¹³⁵Xe γ-ray spectrum . . . 88

5.12 Prompt ¹⁷¹Er γ-ray spectrum . . . 88

5.13 Delayed ¹⁷¹Er γ-ray spectrum . . . 89

5.14 Prompt ¹³⁴Xe γ-ray spectrum . . . 91

5.15 Q-value of the −2n transfer channel . . . 92

5.16 Q-value gated prompt ¹³⁴Xe spectra . . . 92

5.17 Prompt ¹⁷²Er γ-ray spectrum . . . 93

5.18 Delayed ¹⁷²Er γ-ray spectra . . . 94

5.19 Prompt ¹³³Xe γ-ray spectrum . . . 95

5.20 Tentative prompt ¹⁷³Er γ-ray spectrum . . . 96

5.21 Tentative delayed ¹⁷³Er γ-ray spectrum . . . 97

(13)

List of Tables

4.1 Reference position for MCP calibration . . . 41 4.2 Statistics on atomic number Z . . . 49 4.3 Ground state Qgs-values for different reaction channels . 71 4.4 Statistics on validation stages in the analysis procedure . 73 4.5 Known isomers in 68Er isotopes . . . 76

(14)

(15)

1. Introduction

Already in ancient times both Indian and Greek natural philosophers proposed that the physical world around us consists of undividable or uncuttable (ατoµoν) “particles too small to be seen” [1]. These atoms were thought to carry properties or qualities rather than being actual building blocks of matter. Modern atomic theory developed from sci- entific discoveries in the late 18th century. Thomson and Rutherford showed that atoms consist of even smaller particles leading to the Bohr model of the atom: electrons revolve on specific orbits around the dense atomic nucleus, which consists of protons and neutrons. Today’s state of knowledge about elementary particle physics is summarised in the Stan- dard Model. It classifies all known elementary particles and connects them via fundamental force. Yet, the Standard Model has limitations such as how to introduce gravity as a fundamental interaction, which is instead described by general relativity.

The four known forces – weak, strong, electromagnetic and gravita- tional – ultimately define all interactions between elementary particles.

Naïvely, all observable phenomena could be calculated by first principles from theses forces. However, most phenomena are many-body problems and not analytically solvable since N constituents have, for example, N² possible two-body interactions.

Instead, a mean field approach is chosen. It identifies and captures the relevant degrees of freedom for a specific system. The interactions of all constituents with each other are approximated by a system of constituents moving freely in an average (or mean) field. This mean field potential is “generated” by all constituents and corresponds to an effective interaction obtained from experiments or constructed ab initio.

This effective interaction is relevant at the degrees of freedom previously chosen.

The constituents of a finite many-body system can be envisioned as being confined in a volume. The surface of that volume represents the mean field potential of the many-body system. The surface of a finite many- body system may oscillate or even rotate when deformed. An example of such a finite many-body system is the atomic nucleus with proton and neutrons as constituents interacting via the strong and Coulomb force.

Nuclei of intermediate mass have been studied from first principles and recent developments like similarity renormalisation group techniques, effective field theory and many-body methods push the mass frontiers up to Ni and Sn [2].

(16)

Ab initio calculations fail for higher mass numbers A = Z + N with proton number Z and neutron number N. To even simplify the study of lighter nuclear systems, the strong interaction is treated effectively. Dif- ferent models depending on phenomenological observations are available.

To this day, open questions about the nature of the strong interaction at nucleon sizes remain: How many constituents are allowed? Which N/Z-imbalance is allowed? How do nucleon-nucleon (or higher order) interactions lead to the observed phenomena in nuclear matter?

This work aims at studying neutron-rich rare-earth nuclei close to midshell Dy and their yrast structure. At Instituto Nazionale di Fisica Nucleare - Laboratore Nazionali di Legnaro (INFN-LNL)a ¹³⁶Xe-beam is prepared, impinging on a¹⁷⁰Er-target to populate high-spin states in the region of¹⁷⁰Dy. Theheavy-ion magnetic spectrometer with large acceptance angle (PRISMA)identifies beam-like reaction products. Their target- like partners are calculated with the binary partner method. But due to neutron evaporation, additional methods are needed to identify target- like reaction products. TheAdvanced GAmma Tracking Array (AGATA) spectrometer measures γ-ray energies at high efficiencies. The novel techniques of pulse-shape analysis (PSA) and tracking enable Doppler correction of measured γ-ray energies. Three multi-channel plate DANTE detectors surrounding the target acquire kinematic information on reaction products for improved Doppler correction in coincidence withAGATA.

1.1 Outline of this Thesis

This licentiate consists of six chapters. The introduction in this chapter 1 is followed by the theory of deformed nuclei in chapter 2. Chapter 3 gives an overview of the experimental setup. The data analysis is presented in detail in chapter4and I have performed all steps explained there. The consistent representation of the ion trajectory reconstruction in linear algebra in section 4.2.5 is performed by myself. Joa Ljungvall contributed with ideas on the calibration of the ionisation chamber. The good mass resolution was partly obtained following a new calibration method by Marco Siciliano. Some of the empirical corrections described in this chapter evolved from discussions with Philipp John, Benedikt Birkenbach and Marco Siciliano. The results are discussed in chapter5 and concluding remarks can be found in chapter6.

(17)

2. Structure of Deformed Nuclei

The angular distribution and/or specific energies of particle and γ-ray emissions from an excited nucleus give hints on its structure. In the nuclear shell model, each nucleon moves independently in a spherical potential created by all the nucleons. This approach explains the partic- ularly stable nuclear configurations with a sharp discontinuity for proton and neutron separation energies at certain proton and neutron numbers.

These magic numbers correspond to closed shells for both protons and neutrons and give rise to many stable isotopes at proton and isotones at neutron shell closures. Excited states correspond to rearrangements in shell occupation. Nucleons outside of closed shells determine the nuclear structure properties.

This description breaks down when more and more valence nucleons outside closed shells are added, allowing for more and more rearrangements in shell occupation. Instead, collective degrees of freedom govern the structure of the nucleus [3]: macroscopic models describe excitation modes of deformed nuclei where the nuclear shape defines the degrees of freedom. In contrast, microscopic models rely on the motion of particles in a deformed potential.

The details of the following considerations can be found in [4,5].

2.1 Rotational Model

In the macroscopic description all nucleons form a compact entity with the only possible motion being rotation. Compared to a rotating rigid body in classical mechanics, a quantum mechanical axial symmetrical system cannot rotate around its symmetry axis¹. Three “Euler angles”

connect the laboratory system (x, y, z) to the intrinsic system (x⁰, y⁰, z⁰) of the rigid body at rest and axial symmetry along z⁰: two angles determine the direction of the symmetry axis z⁰ in the laboratory frame (x, y, z) and the third angle the orientation of the rigid body along its symmetry axis z⁰. Both coordinate systems share the same point of origin.

1Axial symmetry implies that the wave function of the system does not change under rotation.

(18)

The rotational angular momentum operator Jrot and the intrinsic angular momentum operator J⁰ yield the total angular momentum J of a rotating rigid quantum body

J = J_rot+ J⁰. (2.1)

The intrinsic system is axially symmetric around z⁰resulting in the conservation of the component Jz⁰ of the angular momentum. It is usually denoted as quantum number K. In the laboratory system, however, the direction of rotation is indistinguishable or spherically symmetric. A spherical symmetry leads to the conservation of the total angular momentum J . This implies that J² with eigenvalues J(J + 1) and J^z usually denoted by M are good quantum numbers.

The Hamiltonian H of a rotating rigid quantum body consists of an intrinsic part Hintr. (due to the intrinsic motion of all nucleons inside a deformed potential) and a rotational part H^rot

H = Hintr.⁰ +XJrot,i²

2Iⁱ

(2.1)

=Hintr.⁰ +X(Jⁱ− Ji⁰)² 2Ii

=Hintr.⁰ +XJi⁰²

2Iⁱ

| {z }

Hintr.

−XJiJi⁰

Iⁱ

| {z }

HCoriolis

+XJi²

2Iⁱ

| {z }

Hrot

(2.2)

with the moment of inertia I along the three different axes i. The middle term describes the coupling between intrinsic and rotational degrees of freedom as compared to the classical Coriolis interaction. It can be neglected as long as rotational energies are small compared to intrinsic energies. At higher rotational energies it lowers the observed energy levels – known as backbending [6] – by breaking up paired nucleons, which align their angular momenta to the rotation axis instead of coupling to spin 0. Centrifugal stretching is neglected in the above description.

The Hamiltonian Hrot given by equation (2.2) simplifies for an axial symmetrical nucleus with moments of inertia I^x = Iy = I and no rotational degree of freedom around the symmetry axis z⁰ to

Hrot= Jx²⁰+Jy²⁰

2I = J²− Jz²⁰

2I . (2.3)

The energy eigenvalues E of the Hamiltonian Hrot, defined in equation (2.3) above for deformed nuclei, are given by

E(J, K) = 1

2I J(J + 1) − K² . (2.4)

(19)

For fixed K, energy levels with energies E ∝ J(J + 1) arise, the so- called rotational bands. The lowest lying rotational band corresponds to K = 0with the ground state E(0, 0) = 0 for even-even deformed nuclei and energies

E(J, 0) = J (J + 1)

6 E(2, 0)

for higher lying states. The energy of the first 2⁺1 state is denoted by E(2, 0). Since the Hamiltonian conserves parity, the ground state rotational band is characterised by even J and parity Π = (−1)^J = +1. The quantum number K has to be even for K > 0 and the allowed eigenvalues for the angular momentum operator are given by J = K, K + 1, K + 2.

For even-even nuclei, the first excited 2⁺1 state decays to the 0⁺ground state by electric quadrupole, E2, radiation due to parity conservation and since the difference in angular momentum between ground and first excited state is ∆J = 2. The strength (or probability) of this γ-transition is referred to as “reduced transition probability B(E2)”. For in-band transitions (∆K = 0), the reduced transition probability B(E2) is given by

B(E2 : Ji→ Jf) = 5

16πe²Q²₀hJiKi ∆J = 2∆K = 0|JfKfi² with a Clebsch-Gordan coefficient connecting the initial Ji and final Jf

state. Measuring the reduced transition probability B(E2) allows to deduce the intrinsic quadrupole moment Q0 of a nucleus. Thus, it is a measure of the deformation of nuclei. In the vicinity of closed shells B(E2) values are small, usually in the order of a few Weisskopf units (1W.u. = 5.94 · 10⁻⁶A^4/3e²b²). However, rare-earth or actinide nuclei have B(E2) values of several hundreds of Weisskopf units clearly indi- cating strong nuclear deformation and large collectivity.

The moment of inertia I of an even-even rotational nucleus is obtained experimentally by measuring the energy E(J, 0) and rearranging equation (2.4). It appears to lie in between the moment of inertia for a fluid and a rigid body

Ifluid<Inuclear<Irigid

suggesting that nuclear matter behaves as a superfluid due to pairing.

The rotational model is successfully explaining the γ-ray spectra of a wide range of deformed nuclei.

2.2 Deformed Shell Model

There are many approaches to understand the microscopic nature of the rotational model like the Cranking model [7] or Nilsson model [8]. To

(20)

resemble the residual interactions between many valence nucleons, these models break the spherical symmetry of the shell model potential. In- stead, a deformed potential, for example an axial symmetrical potential, is considered in which orbital angular momentum l and intrinsic spin s are no longer conserved. Thus, nuclear states are rearranged and their energies depend on the projection Ω of the single-particle orbital angular momentum j on the symmetry axis. Due to the axial symmetry, each state is doubly degenerated.

2.3 Surface Vibrations

Phenomenologically, nuclear matter is treated as a liquid droplet of radius R0. Within this model, a possible excitation mode can be visualised as a vibrating nuclear surface. Each point on that surface is described by a time-dependent expansion of the radius R of the nuclear volume in spherical harmonics Yλ^µ

R(ϑ, ϕ, t) = R0



1 +

∞

X

λ=0 λ

X

µ=−λ

αλµ(t)Y_λ^µ(ϑ, ϕ)



. (2.5) The spherical tensor α(t) denotes the time-dependent expansion coefficient, the order λ of the spherical harmonics Y represents the order of excitation and different vibration modes are referred to by µ. At the zeroth order of excitation (λ = 0), the spherical nucleus oscillates in a

“breathing mode” [9]. Due to the incompressible nature of nuclear matter, this monopole mode is strongly prohibited and occurs only at high excitation energies. The dipole λ = 1 mode corresponds to a translation.

Interesting oscillation modes are quadrupole λ = 2, octupole λ = 3 and hexadecapole λ = 4 vibrations.

In the case of quadrupole vibrations, the five different expansion coefficients α2µ are replaced by three Euler angles and two intrinsic parameters. The orientation of the nucleus is fixed by the Euler angles such that the principal axes of the nucleus coincide with the coordinate axes.

The two intrinsic parameters quantify the nuclear shape distortion and are further substituted by the deformation parameter β and symmetry parameter γ

α20= a0= β cos γ a2= 1

2β sin γ.

This change of notation from the expansion coefficients α in the laboratory frame of reference to the parameters a in the intrinsic frame is referred to as the “Lund convention”. The norm of the expansion coeffi-

(21)

cients is rotationally invariant X

µ

kα^µk²= a²₀+ 2a²₂= β². (2.6) From the equation (2.6) it follows directly that high absolute values of β correspond to large deformations. A spherical shape is characterised by β = 0, prolate deformation by β > 0 and oblate deformation by β < 0.

An axially symmetric nucleus identifies as γ = 0 and γ 6= 0 denotes a nucleus with three differently long semi-axes, a so-called triaxial nucleus.

2.4 Multi-Nucleon Transfer Reactions

Multi-nucleon transfer reactions are a type of binary, fast and direct deep- inelastic scattering reaction [10, 11, 12] above the Coulomb barrier. In such reactions, the reaction partners (beam and target) transfer several nucleons between each other. The nuclear surfaces of the two partners are in contact at grazing distance, allowing for a fast redistribution of protons p and neutrons n among the nuclei. However, due to an immediate separation, the rearrangement within the nuclei is minimal, leaving the reaction products in beam- and target-like states. Moreover, angular momentum from the relative orbital motion of the reaction partners is to some extent transferred as intrinsic spin to the reaction products.

Thus, multi-nucleon transfer reactions create the reaction products in beam- and target-like high-spin states – see figure2.1.

The transfer of nucleons is described relative to a beam nucleus. For example, the +1n/ − 1n channel means that a neutron has been transferred to/from a beam from/to a target nucleus. This convention is used throughout the text.

Multi-nucleon transfer reactions are an excellent method to study neutron-rich rare-earth nuclei at high spin states. It is possible to tune in these reactions to a region in populated reaction products of interest.

Binary reactions above the Coulomb barrier have a tendency to equalize neutron over proton number ratios N/Z between the two reaction partners [13]. In this work, a ¹³⁶Xe-beam hits a ¹⁷⁰Er-target with neutron over proton number ratios

N Z ₁₃₆

Xe

= 82

54 ≈ 1.52 and N

Z ₁₇₀

Er

= 102 68 = 1.5

that slightly favour neutron transfer from ¹³⁶Xe nuclei to ¹⁷⁰Er nuclei and vice versa in the case of proton transfer.

Multi-nucleon transfer reactions are described for example by the GRAZINGmodel [14], in which the angular dependency of the cross-section

(22)

n pTargetnucleus¹⁷⁰Er Beamnucleus¹³⁶Xe Multi-nucleon transfer Beam-like Target-like Particle evaporation De-excitation

ITL~ IBL~

170Er, 1 mg/cm²

136Xe, E = 850 keV, 2pnA, q = 28⁺ Coil Calibration cross GridCarbon foil

MCP

1

~ITL

~IBL

170Er, 1 mg/cm²

MCP

1

~ITL

~IBL

170Er, 1 mg/cm²

MCP

1

~ITL

~IBL

170Er, 1 mg/cm²

MCP

1

~ITL

~IBL

170Er, 1 mg/cm²

MCP

1

~ITL

~IBL

170Er, 1 mg/cm²

MCP

1

~ITL

~IBL

170Er, 1 mg/cm²

MCP

1

~ITL

~IBL

170Er, 1 mg/cm²

MCP

1

~ITL

~IBL

170Er, 1 mg/cm²

MCP

1

~ITL

~IBL

170Er, 1 mg/cm²

MCP

1

~ITL

~IBL

170Er, 1 mg/cm²

MCP

1

~ITL

~IBL

170Er, 1 mg/cm²

MCP

1

~ITL

~IBL

170Er, 1 mg/cm²

MCP

1

~ITL

~IBL

170Er, 1 mg/cm²

MCP

1

~ITL

~IBL

170Er, 1 mg/cm²

MCP

1

ITL~ IBL~

170Er, 1 mg/cm²

MCP

1

ITL~ IBL~

170Er, 1 mg/cm²

MCP

1

~ITL

~IBL

170Er, 1 mg/cm²

MCP

1

~ITL

~IBL

170Er, 1 mg/cm²

MCP

1

ITL~ IBL~

170Er, 1 mg/cm²

MCP

1

ITL~ IBL~

170Er, 1 mg/cm²

MCP

1

ITL~ IBL~

170Er, 1 mg/cm²

MCP

1

~ITL

~IBL

170Er, 1 mg/cm²

MCP

1

~ITL

~IBL

170Er, 1 mg/cm²

MCP

1

ITL~ IBL~

170Er, 1 mg/cm²

MCP

1

np Targetnu

cleus ₁₇₀ Er Beam

nucleus₁₃₆ Xe Multi-n

ucleon transf

er Beam

-like T arget-l

ike Particle

evapor ation De-excit

ation ITL~

IBL~

170Er , 1m

g/cm

2 136Xe

, E= 850keV, 2 pnA

, q= 28+

CoilCalibra tioncro

ss Grid

Carbon foil MCP

1

npTargetnucleus

170Er

Beamnucleus

136Xe

Multi-nucleontransfer Beam-likeTarget-like

Particleevaporation De-excitation

~ITL

~IBL

170Er, 1mg/cm

2

136Xe, E

=850 keV,

2 pnA, q=28

+

Coil Calibration cross GridCarbon foil

MCP

1

Beamnucleus

136Xe, E = 859 MeV, 2pnA, q = 28⁺ RD= 1.2 m AGATA Detect E , ~x Doppler correction for BLFs and TLFs Prompt and delayed

rays for TLFs Targetnucleus

170Er, 1 mg/cm², 97.9%enriched

PRISMA @ 44 grazing angle Identify BLFs: Z, A, q, T OF, ~v Kinematics of TLF via binary partner calculation

136Xe Beam @ 859 MeV

1

Figure 2.1. Abeamandtargetnucleus at grazing distance transfer nucleons between each other. Their relative orbital motion contributes to some extend to the intrinsic spin of beam- and target-like reaction products. These products de-excite via particle and γ-ray evaporation.

for different transfer channels is calculated. The angle at the maximum of the cross section is called the “grazing angle”. Reaction partners observed at that angle are in closest contact.

Reaction fragments, produced in a wide range of transfer channels, excitation energies and spin states, de-excite via particle and γ-ray emission.

To explore their yrast structure, high-efficiency γ-ray spectrometers with spatial resolution allowing for Doppler correction of γ-ray energies are deployed. To disentangle the observed γ-ray energies, an unambiguous event-by-event identification of at least one reaction product is needed using a precise mass spectrometer. Information on the other reaction product can be recovered via two-body kinematics. Neutron evaporation complicates the identification in case of neutron-rich nuclei.

2.5 Recent Findings in Neutron-Rich Rare-Earth Nuclei

Neutron-rich rare-earth nuclei lie in the middle of proton and neutron shell closures at Z = 50, 82 and N = 82, 126. Due to their large number of valence nucleons outside closed shells, they represent a region of pro- nounced quadrupole collectivity. Collective observables like the energy E₂⁺

1 and the reduced transition probability B(E2 : 0⁺ → 2⁺1)of the first excited 2⁺1 state as well as the energy ratio E4⁺₁/E₂⁺

1 between the first excited 4⁺1 and 2⁺1 states are quite stable over a wide range of nuclei and depend smoothly on the product of valence nucleons NpNn [15,16, 17]. This proton-neutron valence product NpNn captures the competi-

(23)

tion between pairing interaction of alike nucleons and proton-neutron interactions. The latter conditions the onset of deformation via mixing of shell model configurations. The energy ratio E4⁺/E₂⁺ between the first excited 4⁺1 and 2⁺1 states ranges from 2 for a quadrupole vibrator via 2.5 for a non-axial or γ-soft rotor up to 3.33 for an ideal symmetric rotor [16]. The proton-neutron valence product NpNn peaks at the doubly-midshell nucleus ¹⁷⁰Dy, which may be one of the most rigid axially deformed nuclei with A < 208 in its ground state in nature [18].

Nevertheless, microscopic effects and subshell closures [16, 19, 20, 21]

complicate the simple NpNn-picture leaving the maximum of collectivity in the neutron-rich rare-earth region unknown.

In addition, theoretical calculations [18,22] suggest that the high degree of axial symmetry and deformation in the doubly midshell nucleus

170Dy implicate a very robust K = 6⁺ isomeric state with an estimated half life of the order of hours. Such long half lives cannot be measured with the setup used in this experiment. However, other K isomers with half lives in the µs range are predicted in neighbouring nuclei and in reach for the current experiment. Lately, long-lived K isomers have been re- ported over a wide range of nuclei in the neutron-rich rare-earth region [23,24,25,26,27,28].

Recently, yrast states up to spin 6⁺ were populated in ¹⁷⁰Dy following β-decay after in-flight fission and a K = 6⁺ isomeric state at 1643.91(23) keV with a half life of 0.99(4) µs was identified [24]. γ- vibrational mixing reduces the K-hindrance of this isomeric state, thus shortens its half life as compared to prediction.

However, populating these neutron-rich nuclei is experimentally chal- lenging due to the lack of convenient nuclear reactions and neutron evaporation. Multi-nucleon transfer reactions have been used to populate, for example, yrast states up to spin 10⁺ in ¹⁶⁸Dy [29], up to spin 41/2⁻ in

169Er [30], up to spin 45/2⁻ in ¹⁷¹Er [30], up to spin 22⁺ in ¹⁷²Er [30]

and up to spin 14⁺ in ¹⁷⁸Yb [31]. Using decay spectroscopy following in-flight fission yrast states up to spin 8⁺ in¹⁷²Dy [23] were populated.

(24)

3. Experimental Setup

This work presents the analysis of experiment 11.25 performed in October 2011 at INFN-LNL, which is located close to Padova in Italy. A ¹³⁶Xe- beam was delivered by the Tandem-ALPI accelerator complex, hitting a 97.7% enriched ¹⁷⁰Er-target with a thickness of 1 mg/cm² and leading to multi-nucleon transfer reactions. Reaction products were identified withPRISMA, see section3.2.1, and coincident γ rays were detected and tracked by theAGATAspectrometer, see section 3.2.2. Five days of beam time were approved for the experiment.

Figure 3.1. The photographs of the setup were taken during the assembling of this experiment. The beam enters from the left through the red beam tube.

Close to AGATA, which is mounted in a golden support frame, a part of the beam line is removed as well as half of the target chamber. The target chamber can be seen both in the middle of the photograph and in the insert as a silver metallic hemisphere. The hole to the right is the entrance toPRISMAand a ring with three DANTE detectors is visible. The data of the DANTE detectors is not analysed in the present work. The yellow structure to the right is the mounting platform ofPRISMAwith the magnetic optics in blue.

(25)

3.1 LNL Accelerator

In this experiment, the Tandem-ALPI accelerator complex at INFN-LNL was used to provide the ¹³⁶Xe ion beam with an energy of 859 MeV, a beam current¹ of up to 2 pnA and a charge state of q = +28.

The Tandem-XTU is a two-stage Van de Graaff accelerator placed in a horizontal vessel filled with sulfur hexafluoride (SF6) at 7 atm for insu- lation purposes. Weakly positively charged ions are extracted from an external ion source. They enter the low-energy side of the tandem accelerator via a charge changing region where they are weakly negatively charged and accelerated towards the high-voltage terminal with maximum +14.5 V [32] in the center of the vessel. Passing a 5 µg/cm²charge stripping carbon foil, the ions are repelled from the high-voltage terminal.

This second (tandem) acceleration depends on the charge state of the ions and resembles the high-energy side of the accelerator. The Tandem-XTU provides beam energies from 30 MeV/u for¹H to ∼ 1.5 MeV/u for¹⁹⁷Au.

The beam is delivered to different experimental station by switching magnets or injected into the ALPI accelerator to further boost the beam energy.

ALPI(Acceleratore Lineare Per Ioni) is a linear superconducting accelerator for heavy ions. It consists of quarter wave resonance (QWR) Cu-Pb or Cu-Nb cavities grouped into low-, medium- and high-β sec- tions [33]. Four cavities share a common cryostat cooled with liquid helium to 4.2 K keeping the cavities in a superconducting regime. With a pulsing system [34] the beam delivered by the Tandem-XTU is bunched at 80 MHz and in phase with the radio frequency (RF) electromagnetic waves coupled into the cavities in the low-β section. The RF creates an electric field with maximum 4.4 MV/m bunch acceleration. ALPI boosts ion beams from ²⁸Si to¹⁹⁷Au to a maximum energy of about 35 MeV/u.

Switching magnets deliver the ion beam to the same experimental sta- tions as the tandem accelerator.

3.2 Detector Systems

In this experiment,PRISMAwas placed at the grazing angle of 44^◦[14,35]

and used to identify the atomic number Z, the mass number A as well as the velocity v and kinetic energy E of beam-like reaction products.

The AGATA spectrometer was placed opposite to the entrance to PRISMA – as can be seen in figure 3.1 – and tracks γ rays in coincidence with the beam-like ions. Six LaBr3 fast-timing detectors surroundAGATA and three DANTE detectors [36, 37] were placed around the target. In the present work, only the data fromPRISMAand AGATAis analysed.

1The unit “particle nano Ampère” is defined as 1 pnA = 6.25 · 10⁹ ions/s.

Collectivity in Neutron-Rich Erbium Isotopes

Aila Gengelbach