Finally, I would like to express my heartfelt gratitude to my parents for their love, care and constant

First and foremost, I would like to express my sincere gratitude to my supervisor Professor Costel Petrache, for his systematic teaching and training me in experimental nuclear structure research. It is my great honor to have a supervisor who is so famous and highly respected in the nuclear physics. Also, I want to express my deepest grateful to him for his take care of me, and all the things he has done for me even if he never told me. In addition, for sure, his personal qualities, like work hard, great passion for research, explorative spirit ......, will influence me more in the future. For me, he is not only a mentor in my work, but also in my life.

I would like to also thank my CSNSM colleagues. Alain Astier help will never be forgotten due to his selfless shared many data analysis skills used in present work. I feel very lucky that I had a very nice officemate, Etienne Dupont, who assisted me a lot in my daily life in France. I want to say "Merci beaucoup, monsieur Etienne Dupont ". I also would like to thank Amel Korichi, Araceli Lopez-Martens, Jérémie Jacob, Nicolas Dosme and all the other colleagues in CSNSM for their assistance.

I also thank the nuclear structure group of IMP, Lanzhou, in particular, Xiaohong Zhou, my supervisor, also Guo Song and Jianguo Wang for initiating me in experimental nuclear physics research. In particular, I would like to thank Dr. Guo Song for establishing the collaboration between Professor Costel Petrache group and the nuclear structure group in IMP, which provided me the opportunity to do my PhD in CSNSM, Orsay, France. I also would like to thank another one of my supervisors, Wenhui Long from Lanzhou University for his continue support and very wisdom advice for my future career. I would like to thank all our collaborators, in particular to Prof. Meng Jie and Dr. Qibo Chen for their excellent theoretical work made for the interpretation of the present data.

I am thankful to the jury members of my PhD defense, Dr. Amel Korichi (CSNSM, Orsay), Dr. Elena Lawrie (iThemba Labs, South Africa), Dr. Nadine Redon (IP2I Lyon), and Dr. Zhong Liu (IMP, Lanzhou) for having spent their valuable time to evaluate my PhD work and read my thesis, also to give suggestions and comments on my manuscript.

CONTENTS

List of Figures

1.1 Left-and right-handed chiral systems for a triaxial odd-odd nucleus. The symbols J, R, j⌫ , and j⇡ denote respectively the total angular momentum, the angular momenta of the core, of the neutron and of the proton, respectively. Figure adopted from Ref. [ [START_REF] James | Spectroscopy of Exotic Proton-Rich Nuclei Close to A = 120[END_REF]. . . . . . . . . . . . . . . . . . . . . . . 2.5 Schematic of the quantum numbers which can describe the deformed nucleus. ⇤, ⌦, ⌃, and K are the projections of the orbital angular momentum l, of the total angular momentum of the particle j, of the spin of the particle s, and of the total angular momentum J onto the symmetry axis, respectively. In addition, R is the angular momentum of the core and M is the projection of the total angular momentum onto the laboratory axis. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.6 Nilsson diagram for protons in the 50  Z  80 region showing the single-particle energies as a function of the deformation parameter ✏ 2 . For ✏ 2 >0, corresponding to the prolate shape; for ✏ 2 =0, corresponding to the spherical shape; for ✏ 2 < 0, corresponding to the prolate shape. Labels obey the ⌦ ⇡ [Nn z ⇤] rule. 10 LIST OF FIGURES 2.7 Nilsson diagram for neutrons in the 50  N  80 region showing the single-particle energies as a function of the deformation parameter ✏ 2 . For ✏ 2 >0, corresponding to the prolate shape; for ✏ 2 =0, corresponding to the spherical shape; for ✏ 2 < 0, corresponding to the prolate shape. Labels obey the ⌦ ⇡ [Nn z ⇤] rule. 6.3 (Color online) Comparison between the experimental excitation energies relative to a reference rotor (symbols) and the particlerotor model calculations (lines) for the bands D1-D6. . . . . . 6.4 (Color online) The experimental quasi-particle alignments for the chiral rotational bands of 135 Nd. . . . . . . . . . . . . . . . 6.5 (Color online) Comparison between experimental ratios of transitions probabilities B(M 1)/B(E2) (symbols) and the particlerotor calculations (lines) for the bands D1-D6. . . . . . . . . . 6.6 (Color online) The root mean square components along the intermediate (i , squares), short (s , circles) and long (l , triangles) axes of the rotor, valence protons, and valence neutrons angular momenta calculated as functions of spin by PRM for the doublet bands D3 and D4 of 135 Nd. . . . . . . . . . . . .

7.1

-Time matrix for the clovers at the focal plane. The transitions marked with asterisks represent the -decay contaminants from the nuclei produced in this experiment: 665 keV, 783 keV, 828 keV and 872 from the -decay of 135 Ce, 761 keV and 925 keV from the -decay of 137

List of Tables

5.1

Experimental information including the -ray energies, energies of the initial levels E i , intensities I , anisotropies R DCO and or R ac , multipolarities, and the spin-parity assignments to the observed states in 136 Nd. The transitions listed with increasing energy are grouped in bands. The deduced values for R DCO with a stretched quadrupole gate are ⇡ 1 for stretched quadrupole and ⇡ 0.46 for dipole transitions, while the ratio is close to 1 for a dipole and 2.1 for a quadrupole transition when the gate is set on a dipole transition. The R ac values for stretched dipole and quadrupole transitions are ⇡ 0.8 and ⇡ 1.4. . . . . . . .

5.2

Experimental information including the ray energies, energies of the initial levels E i and the tentative spin-parity assignments to the observed states in 136 Nd. . . . . . . . . . . . . . . . . .

Unpaired nucleon configurations labeled

A-H and the corresponding parities, calculated by constrained CDFT. The excitation energies E x (unit MeV) and quadrupole deformation parameters ( , ) are also presented. . . . . . . . . . . . . . .

5.4

The parities, unpaired nucleon configurations, quadrupole deformation parameters ( , ), moments of inertia J 0 (unit h2 /MeV), and Coriolis attenuation factors ⇠ used in the PRM calculations for bands D1-D6 and their partners. . . . . . . . . . . . . . . .

Configuration assignments and deformation information

Chapter 1 Introduction

The atomic nucleus is a strongly interacting quantum many-body system composed of protons and neutrons. The nuclei can contain nucleons from a few to several hundred. This finite number of nucleons makes the nucleus become a unique system which cannot be treated in a statistical way, and its behaviour at a phase change is very different from that of ordinary matter. The finiteness of the nucleus is also manifest in the influence that just one nucleon may have on determining the nuclear properties, particularly the nuclear shape. The main aim of the nuclear structure is to understand the distribution and motion of the nucleons inside the nucleus and also to study the collective motion and shape of the whole nucleus.

The work presented in this thesis is centred around two nuclei: 135 Nd and 136 Nd, which are located in the A ⇡ 130 mass region, a fertile field of study of the transitional nuclei with shapes that can vary between strongly deformed in the middle of the major shell 50 < Z, N < 82 to nearly spherical close to the shell closure [START_REF] Möller | Global calculations of ground-state axial shape asymmetry of nuclei[END_REF]. A variety of shapes can also be present in a single nucleus in different spin ranges. At low spins, axial asymmetry was suggested for the ground states of nuclei centered around Z = 62, N = 76 [START_REF] Möller | Global calculations of ground-state axial shape asymmetry of nuclei[END_REF], in which a regular increase of the level energies of the observed low-lying band, considered a sign of rigid triaxiality has been observed, while in the surrounding nuclei the level energies present a staggering, which is considered a sign of soft triaxial shapes.

At medium spins, the shape can change under the polarizing effect of unpaired nucleons resulting from broken pairs. In certain cases the triaxial shape becomes more rigid, being based on a deeper minimum of the potential energy surface, induced by the protons occupying low-⌦ orbitals in the lower part of the h 11/2 subshell, or by neutrons excited from orbitals below N = 82 to low-⌦ (h 9/2 , f 7/2 ) orbitals lying above N = 82 (see, e.g., [START_REF] Petrache | Multiple triaxial bands in 138 Nd[END_REF][START_REF] Petrache | Triaxiality and exotic rotations at high spins in 134 Ce[END_REF] and references therein).

At high spins, the nuclei with several holes in the N = 82 shell closure, a multitude of triaxial bands have been observed in several Ce and Nd nuclei, giving strong support for the existence of stable triaxial shape up to very high 18 1.1. Chirality spins. In addition, nuclei of the A ⇡ 130 mass region can also present other coexisting shapes, e.g., axially deformed, highly deformed, and even superdeformed at high spins. This is the case for 140 Nd, in which states based on a spherical shape have been observed up to spins as high as 27 h coexisting with triaxial shapes, and in which axial superdeformed and highly deformed shapes coexist at very high spin [START_REF] Petrache | Search for the terminating 27 state in 140 Nd[END_REF][START_REF] Neusser | Superdeformed band at very high spin in 140 Nd[END_REF]. In this nucleus, the bridge between the regions of highly deformed bands present in A ⇡ 130 nuclei and the superdeformed bands present in A ⇡150 nuclei provides an insight into the development of the deformation between these two regions of superdeformation [START_REF] Neusser | Superdeformed band at very high spin in 140 Nd[END_REF].

An additional feature of nuclei in this mass region is the presence of isomeric states, e.g., the 10 + , T 1/2 = 370 ns isomer of 138 Nd, the 10 + , T 1/2 = 308 ns isomer of 134 Ce, the 11/2 , T 1/2 = 2.7 µs isomer of 137 Pr, and the 6 + , T 1/2 = 90 ns isomer of 136 Pr [START_REF]ENSDF database[END_REF], populated in the reaction used in the present work, which interrupts and fragments the decay flux.

The A ⇡ 130 mass region is also characterized by the existence of two unique fingerprints of a triaxial nuclear shape: chirality and wobbling. Chirality represents a novel feature of triaxial nuclei rotating around an axis which lies outside the three principal planes of the triaxial ellipsoidal shape, which will be discussed in detail in Section 1.1. The wobbling motion has been discussed by Bohr and Mottelson in the triaxial even-even nuclei many years ago [16]. This mode represents the quantized oscillations of the principal axes of an asymmetric top relative to the space-fixed angular momentum vector or, in the body fixed frame of reference, the oscillations of the angular momentum vector about the axis of the largest moment of inertia [START_REF] Matta | Transverse wobbling in 135 Pr[END_REF]. Recently, the wobbling interpretation has been proposed for low-lying bands observed in 135 Pr [START_REF] Matta | Transverse wobbling in 135 Pr[END_REF].

Thus, this region of the nuclear chart is an ideal testing ground to investigate the competition between various deformations and their evolution with spins, as well as the competition between single-particle and collective modes of excitation.

Chirality

The term chirality, was first time introduced by Lord Kelvin in 1904 in his Baltimore Lectures [START_REF] Thomson | Baltimore lectures on molecular dynamics and the wave theory of light[END_REF]: "I call any geometrical figure, or group of points, chiral, and say it has chirality, if its image in a plane mirror, ideally realized, can not be brought to coincide with itself."

Chirality commonly exists in nature, and has important consequences in fields of science as diverse as biology, chemistry, and physics. The best known examples of geometrically chiral objects are the human hands and the microscopic handedness of certain molecules. Chiral symmetry is also well known in particle physics, where it is of a dynamic nature distinguishing between the two possible orientations of the intrinsic spin with respect to the momentum of the particle [START_REF] Jie | Relativistic density functional for nuclear structure[END_REF].

Nuclear Chirality

In nuclear physics, chirality was suggested in 1997 by Frauendorf and Meng [START_REF] Frauendorf | Tilted rotation of triaxial nuclei[END_REF]. It shows up in a triaxial nucleus which rotates about an axis out of the three principal planes of the ellipsoidal nuclear shape. Figure 1.1: Left-and right-handed chiral systems for a triaxial odd-odd nucleus. The symbols J, R, j⌫ , and j⇡ denote respectively the total angular momentum, the angular momenta of the core, of the neutron and of the proton, respectively. Figure adopted from Ref. [START_REF] Meng | Open problems in understanding the nuclear chirality[END_REF].

The simplest chiral geometry is expected in an odd-odd nucleus when the angular momenta of the valence proton, of the valence neutron, and of the core tend to be mutually perpendicular. This occurs when the collective angular momentum is oriented along the intermediate axis (this happens if one assumes hydrodynamical moments of inertia), and the angular momentum of the high-j quasiparticle and quasihole are aligned along the short and long axes, respectively, minimizing thus the interaction energy. These three mutually perpendicular angular momenta can be arranged into a left-or a right-handed system (see Fig. 1.1), which differ by intrinsic chirality; the two systems are related by the chiral operator of the form = T R(⇡), where R(⇡) corresponds to a rotation of 180 , while T is the time reversal and therefore changes to opposite the directions of all angular momentum vectors. Chiral symmetry in an atomic nucleus is observed because quantum tunneling occurs between systems with opposite chirality. When chiral symmetry is thus broken in the body-fixed frame, the restoration of the symmetry in the laboratory frame is manifest as degenerate doublet I = 1 bands with the same parity, called chiral doublet bands [START_REF] Starosta | Chiral doublet structures in odd-odd N = 75 isotones: Chiral vibrations[END_REF][START_REF] Vaman | Chiral degeneracy in triaxial 104 Rh[END_REF]. In practice, chiral bands are near degenerate, and show similar properties. The chiral symmetry can be identified if similarities between a specific band and its partner band are observed. The specific fingerprints used to identify chiral doublet bands will be discussed in Section 1.2.

Chirality

It should be pointed out that the chirality in the nature is often static, like in the molecules, in the DNA. In the case of nucleus, the chirality is dynamic, since it is the angular momentum vector that defines a direction with respect to which the semi-axes (1, 2, 3) form a left-or right-handed system. The non-rotating nuclei are achiral.

Multiple chiral doublets (M D)

Adiabatic and configuration-fixed constrained triaxial relativistic mean field (RMF) approaches were developed to investigate the triaxial shape coexistence and possible chiral doublet bands in 2006 [START_REF] Meng | Possible existence of multiple chiral doublets in 106 Rh[END_REF], which predicted a new phenomenon, the existence of multiple chiral doublets (M D), i.e., more than one pair of chiral doublet bands, in one single nucleus. This phenomenon was suggested for 106 Rh after examining the possible existence of triaxial deformation and of the corresponding high-j proton-hole and neutron-particle configurations. Such investigation has been extended to the rhodium isotopes and the existence of M D has been suggested in 104,106,108,110 Rh [START_REF] Peng | Search for multiple chiral doublets in rhodium isotopes[END_REF]. The investigation predicted M D not only in 106 Rh, but also in other mass regions, i.e., A ⇡ 80, and A ⇡ 130.

Recently, M D bands have been identified in A ⇡ 130, A ⇡ 110, and A ⇡ 80 mass regions. The first experimental evidence for the existence of M D in the A ⇡ 130 mass region was reported in 133 Ce in 2013 [START_REF] Ayangeakaa | Evidence for multiple chiral doublet bands in 133 Ce[END_REF]. It was found that the negative-parity bands 2 and 3, and the positive-parity bands 5 and 6 based on the 3-quasiparticle configurations ⇡[(1h 11/2 ) 1 (1g 103 Rh [START_REF] Kuti | Multiple chiral doublet bands of identical configuration in 103 Rh[END_REF], where an "excited" chiral doublet of a configuration is seen together with the "yrast" one. This observation showed that chiral geometry can be robust against the increase of the intrinsic excitation energy. In 78 Br, two pairs of positive-and negative-parity doublet bands together with eight strong electric dipole transitions linking the yrast positive-and negative-parity bands have been identified [START_REF] Liu | Evidence for octupole correlations in multiple chiral doublet bands[END_REF]. They were interpreted as M D bands with octupole correlations, being the first example of chiral geometry in octupole soft nuclei and indicating that nuclear chirality can be robust against the octupole correlations.

It should be noted that until now all the observed M D bands are only in odd-odd, odd-even, and even-odd nuclei. It inspired us to search for the M D bands in even-even nuclei. This is one of the main aims of the experiment presented in this thesis.

Fingerprints of the chiral bands

From an experimental point of view, the doublet bands must satisfy a set of criteria in order to be recognized as chiral bands, among which the most important are the energy separation between the partners and their electromagnetic transitions rates.

Energy spectra

Firstly, the appearance of near degenerate I = 1 bands with the same parity is considered to be one fingerprint for the chiral bands. The energies of the partner bands should be close to each other, i.e. be nearly degenerate. What does near degenerate mean is not precisely defined since it depends on the deformation, valence nucleon configuration and their couplings. Normally, the acceptance of a near degenerate energy is below 200 keV.

From the measured energies one can derive other observables, like the energy staggering parameter S(I) = [E(I) E(I 2)]/2I which can also serve as fingerprint for chiral partner bands.

Experimentally, the signature splitting between the odd-and even-spins sequences of a given dipole band is quantified by the energy staggering S(I). When the rotation axis is tilted outside the principal planes, the signature is not a good quantum number and therefore it is more appropriate to speak about the odd/even spin dependence of S(I). The expected typical behaviour of S(I) in a chiral band is a small odd/even staggering at low spins, which diminishes with increasing spin, and finally, becomes constant. Therefore, the energy staggering parameter should be almost constant and equal for the states of the same I, in the two chiral partners [START_REF] Aa Raduta | Specific features and symmetries for magnetic and chiral bands in nuclei[END_REF].

Electromagnetic transitions rates

The electromagnetic transitions rates in chiral doublet bands obey important selection rules. Thus, for odd-odd nuclei, with the ⌫h 11/2 ⌦ ⇡h 1 11/2 configuration coupled to a rigid triaxial rotor with = 30 , the selection rules for electromagnetic transitions in the chiral bands has been proposed in Ref. [START_REF] Koike | Chiral bands, dynamical spontaneous symmetry breaking, and the selection rule for electromagnetic transitions in the chiral geometry[END_REF], including the odd-even staggering of intraband B(M 1)/B(E2) ratios, the interband B(M 1) values, as well as the vanishing of the interband B(E2) transitions in the high spin region.

The fingerprints of the electromagnetic transitions probabilities in the chiral bands also depend on the deformation, valence nucleon configuration and their coupling. It is found that the B(M 1) staggering depends strongly on the character of the nuclear chirality, i.e., the staggering is weak in chiral vibration region and strong in the static chirality region. For partner bands the similar B(M 1), B(E2) transitions, and the strong B(M 1) staggering can be used as a fingerprint for the static chiral rotation [START_REF] Qi | Examining B(M1) staggering as a fingerprint for chiral doublet bands[END_REF]. This result agrees 1.2. Fingerprints of the chiral bands with the lifetime measurements for the doublet bands in 128 Cs [START_REF] Grodner | 128 Cs as the best example revealing chiral symmetry breaking[END_REF], but not in 135 Nd which shows a chiral vibration [START_REF] Mukhopadhyay | From chiral vibration to static chirality in 135 Nd[END_REF].

Other fingerprints

To identify chiral doublet bands besides energy spectra and electromagnetic transitions rates, as arguments in favor of chirality there are some other specific fingerprints. Experimentally, almost constant energy difference between partners, and similar moment of inertia are always used in the discussion of the chiral doublet bands. In addition, one can examine the similarities of the configurations for chiral doublet bands by analysing the I-h! relation, where h! is the rotational frequency defined as h! =[E(I + 1) E(I 1)]/2. Generally speaking, the I-h! relation for the yrast band and its partner band should be similar. Furthermore, one can also examine the angular momentum geometries of the observed doublet bands by calculating the expectation values of the angular momentum components of the core, of the valence protons and of the valence neutrons, along the intermediate, short, and long axes.

Experimentally, the chiral phenomenon has been reported in a number of odd-odd and odd-A nuclei in the mass A ⇡ 80, A ⇡ 100, A ⇡ 130, A ⇡ 190 regions; see e.g., Refs. [START_REF] Starosta | Chiral doublet structures in odd-odd N = 75 isotones: Chiral vibrations[END_REF][START_REF] Vaman | Chiral degeneracy in triaxial 104 Rh[END_REF][START_REF] Ayangeakaa | Evidence for multiple chiral doublet bands in 133 Ce[END_REF][START_REF] Kuti | Multiple chiral doublet bands of identical configuration in 103 Rh[END_REF][START_REF] Liu | Evidence for octupole correlations in multiple chiral doublet bands[END_REF][START_REF] Grodner | 128 Cs as the best example revealing chiral symmetry breaking[END_REF][START_REF] Mukhopadhyay | From chiral vibration to static chirality in 135 Nd[END_REF][START_REF] Zhu | A composite chiral pair of rotational bands in the odd-A nucleus 135 Nd[END_REF][START_REF] Joshi | Stability of chiral geometry in the odd-odd rh isotopes: spectroscopy of 106 Rh[END_REF][START_REF] Alcántara-Núñez | Magnetic dipole and electric quadrupole rotational structures and chirality in 105 Rh[END_REF][START_REF] Timar | Experimental evidence for chirality in the odd A nucleus 105 Rh[END_REF][START_REF] Grodner | Partner bands of 126 Cs -first observation of chiral electromagnetic selection rules[END_REF][START_REF] Wang | The first candidate for chiral nuclei in the A ⇡ 80 mass region: 80[END_REF][START_REF] Tonev | Candidates for twin chiral bands in 102 Rh[END_REF]. The distribution of the observed chiral nuclei in the nuclear chart is given in Fig. 1.2.

Motivation of this study

The existence of triaxially deformed nuclei has been the subject of a long standing debate. It appears questionable how well the non-axial shape is stabilized. The intimate mechanism which induces such a behavior needs detailed and accurate investigations, both from the experimental and theoretical points of view. Recently, the wobbling and chirality, as the two unique fingerprints of triaxiality in nuclei, have been intensively studied.

The main part of present work was undertaken with the aim of searching for multiple chiral bands in 136 Nd, a nucleus which is the nearest even-even neighbor of the first chiral candidate 134 Pr [START_REF] Frauendorf | Tilted rotation of triaxial nuclei[END_REF] and the first reported wobbler 135 Pr outside of the A ⇡ 160 region [START_REF] Matta | Transverse wobbling in 135 Pr[END_REF]. In addition, an important effort was devoted to the detailed spectroscopy of 136 Nd since most of the experimental results reported previously were obtained more than twenty years ago with less efficient detector arrays.

The other aim of the present work was focused on a detailed study of the chirality in 135 Nd, to search for multiple chiral bands similar to those observed in the 133 Ce isotone [START_REF] Ayangeakaa | Evidence for multiple chiral doublet bands in 133 Ce[END_REF]. The nucleus 135 Nd is one of the best known examples of chiral vibration [START_REF] Mukhopadhyay | From chiral vibration to static chirality in 135 Nd[END_REF][START_REF] Zhu | A composite chiral pair of rotational bands in the odd-A nucleus 135 Nd[END_REF].

In addition to the study of the chiral bands of 135 Nd and 136 Nd, we devoted an important effort to searching for long-lived isomers in the populated nuclei, in particular in 135 Nd and 136 Nd, using the RITU+GREAT setup and the recoil tagging technique. This effort was motivated by the existence of longlived isomers in 134 Nd, 137 Nd, 138 Nd, 139 Nd, and 140 Nd [START_REF]ENSDF database[END_REF], but not in 135 Nd and 136 Nd.

Outline of thesis

The remaining chapters of this thesis will be organized as follows:

The relevant nuclear models needed to understand the experimental results presented in this work are described in Chapter 2.

Details of the experimental techniques, and the data analysis procedures are outlined in Chapter 3 and Chapter 4, respectively.

The experimental results and discussion of 136 Nd and 135 Nd are reported in Chapter 5 and Chapter 6, respectively.

The search for possible long-lived isomeric states in the populated nuclei, in particular in 136 Nd and 135 Nd are presented in Chapter 7.

The detailed information of JUROGAM II detectors are given in Appendix A.

Outline of thesis

Chapter 2

Theoretical backgroud

In this chapter, some of the nuclear structure models relevant to the present work will be presented. The specific mechanism leading to the chiral mode of excitation will also be briefly discussed.

Liquid drop model

The liquid drop model (LDM) [START_REF] Gamow | Mass defect curve and nuclear constitution[END_REF] of the nucleus was historically the first model proposed to describe the different properties of the nucleus. The idea of considering the nucleus as a liquid drop originally came from considerations about its saturation properties and from the fact that nucleus has a low compressibility and a well defined surface [START_REF] Ring | The nuclear many-body problem[END_REF].

One of the basic properties of a nucleus is its binding energy B(N, Z), where N snd Z are the number of neutrons and protons, respectively. Many attempts were made to reproduce the nuclear binding energy B, but the liquid drop model proposed by Weizsäcker [START_REF] Cf V Weizsäcker | Zur theorie der kernmassen[END_REF] and Bethe [START_REF] Albrecht | Nuclear physics a. stationary states of nuclei[END_REF] is one of the successful ones. The semi-empirical mass formula of the Weizsäcker and Bethe for the nuclear binding energy reads:

B = (Nm n + Zm p m)c 2 = a vol A a surf A 2/3 a coul Z 2 A 1/3 a asym (N Z) 2 A + (A), (2.1)
where m n and m p are the masses of neutron and proton, respectively, and A is the mass number. The five coefficients a vol , a surf , a coul , a asym , and are obtained by fitting the experimental data. The physical meaning of Eq. 2.1 is the following:

-the first term is usually called the volume term, because it is proportional to the mass number A;

-the second term, called the surface term, is proportional to the nuclear surface area A 2/3 ; 25 2.2. The shell model -the third term called the Coulomb term derives from the Coulomb interaction among protons, and is proportional to Z 2 ; -the fourth term called the asymmetry term reflects the fact that nuclear force favour equal numbers of neutrons and protons, or N = Z;

-the last term called the pairing term favours configurations where two identical fermions are paired. It can be rewritten as follows:

(A) = 8 < : a p A 3/4 ,
for even even nuclei; 0, for even odd nuclei;

a p A 3/4 ,
for odd odd nuclei.

where a p ⇡ 34 MeV [START_REF] Ragnarsson | Shapes and shells in nuclear structure[END_REF].

The LDM was very successful in predicting the nuclear binding energy and describing how a nucleus can deform and undergo fission. As a collective model, it is also particularly useful in describing the macroscopic behaviour of the nucleus. However, this is a crude model that does not explain all the properties of the nucleus and nuclear shell structure. In particular, it fails when it is used to explain the nuclei with magic numbers (N or Z equal 2, 8, 20, 28, 50, 82, and 126). A more sophisticated model must to be developed to solve this problem.

The shell model

As mention above, one of most important information on the shell structure is the presence of magic numbers. If one of the proton or neutron numbers is equal to a magic number, then the nucleus is more stable compared with the neighbors, has a larger total binding energy, a much larger energy of the first excited state, and a larger energy required to separate one nucleon.

The shell model was firstly proposed by Mayer and Jensen [START_REF] Goeppert | On closed shells in nuclei[END_REF][START_REF] Haxel | On the "magic numbers" in nuclear structure[END_REF] in 1949 to interpret the observed shell structure in nuclei. The basic assumptions of shell model is that each nucleon moves independently in an average potential created by other nucleons. The assumption made is that motion of the nucleons is quite similar to the motion of electrons in an atom. The Schrödinger equation for a given potential V (r), is written as

( h2 2m r 2 + V (r)) i (r) = ✏ i i (r), (2.2)
where i (r) and ✏ i are the eigenstates and eigenvalues representing the wave functions and energies, respectively. Various potential wells have been used, for example square well, infinite harmonic oscillator well, and Woods-Saxon potential.

Harmonic oscillator potential

The harmonic oscillator potential has the form

V (r) = 1 2 m! 2 r 2 , (2.3)
where m is the mass of the nucleon and ! is the angular frequency of the oscillator. The Schrödinger equation of motion for harmonic oscillator is written as:

( h2 2m r 2 + 1 2 m! 2 r 2 ) i (r) = ✏ i i (r). (2.4)
The energy eigenvalues are:

✏ N = h!(N + 3 2 ), (2.5)
where

N = 2(n 1) + l, (2.6)
with n= 1, 2, 3, ..., and l =0, 1, 2,...n 1. N is the principal quantum number, n is the radial quantum number. l is angular momentum, and it often referred to using the spectroscopic notations, s, p, d, f, g, h, ... corresponding to l = 0, 1, 2, 3, 4, 5, .... In this case, all levels with the same principal quantum number N are degenerate, where the degeneracy is given by 2(2l + 1). The parity of each level is given by ⇡ = ( 1) N . This potential can only reproduce the magic numbers 2, 8 and 20, implying that the model needs to be modified if one wants to reproduce higher magic numbers.

Woods-Saxon potential

To improve the model, the more realistic Woods-Saxon potential [START_REF] Roger | Diffuse surface optical model for nucleon-nuclei scattering[END_REF] can also be employed. It has the form

V (r) = V 0 1 + e (r R)/a , (2.7)
where R = r 0 A 1/3 is the mean radius of the nucleus, V 0 ' 50 [MeV] is the depth of the potential well, and a describes the diffuseness of the nuclear surface, a ' 0.5 [fm].

Compared with the harmonic oscillator potential it removes the l degeneracies of the major shells, filling the shells in order with 2(2l+1) levels, but also in this case only the magic numbers 2, 8 and 20 are reproduced. In addition, its eigenfunctions can not be solved analytically, and must be treated numerically.

The deformed shell model

Spin orbit coupling

To reproduce all the magic numbers, a quite different suggestion was put forward independently by Mayer [START_REF] Goeppert | On closed shells in nuclei[END_REF] and by Haxel, Jensen and Suess [START_REF] Haxel | On the "magic numbers" in nuclear structure[END_REF]. They added a strong spin-orbit coupling term l • s into the single-particle Hamiltonian.

The addition of this term splits the states with the same orbital angular momentum l into two. The single particle total angular momentum is defined by j = l + s, and then the j values of the split-up levels are j = l ± 1 2 . By including the spin-orbit coupling term all magic numbers are successfully reproduced, see Fig. 2.1. Other properties, like the spins and parities of the ground states of most spherical nuclei, and the ground state magnetic moments of nuclei are also well described. But the shell model with the spin-orbit coupling term is still not accurate for the description of nuclei far away from the closed shells. These difficulties were overcome by Nilsson who introduced the deformed shell model [START_REF] Gösta | Binding states of individual nucleons in strongly deformed nuclei[END_REF].

The deformed shell model

Deformed parameters

The nucleus is considered as an incompressible liquid drop with a sharp surface or surface oscillations. In oder to investigate these oscillations, we can parametrize them in some way. The length of the radius vector R(✓, ) pointing from the origin to the surface can be written as

R(✓, ) = R 0 [1 + 1 X =0 X µ= ↵ µ Y µ (✓, )], (2.8)
where R 0 is the radius at the spherical equilibrium with the same volume, Y µ are the spherical harmonics. Each spherical harmonic component will have an amplitude ↵ µ , where represents different modes of deformations. The general expansion of the nuclear surface in Eq. 2.8 allows to describe different shapes, as displayed in Fig. 2.2.

The lowest order multipole term, = 1, does not correspond to a deformation of the nucleus, but rather to a shift of the position of the center of mass. Thus, the deformation of order = 1 is equivalent to a translation of the nucleus that should be neglected for nuclear excitations.

The = 2 multipole term corresponds to the quadrupole deformation which looks like ellipsoidal deformation. In this case, the nucleus shape could be oblate, prolate or triaxial. It turns out to be the most important mode of excitation of the nucleus, therefore only the = 2 will be considered in the following discussion. The octupole deformation corresponding to = 3, is the principal asymmetric mode of deformation of the nucleus and is associated with a pear shape.

The term = 4 (hexadecapole) and higher order terms are small and often can be neglected.

In the case of quadrupole deformation, there are five shape parameters ↵ µ ; three of these shape parameters are equivalent to classical Euler angles,

↵ 20 = 2 cos , ↵ 22 = ↵ 2 2 = 1 p 2 2 sin , (2.9)
which satisfy the condition,

X µ |↵ 2µ | = ↵ 2 2 2 + 2↵ 2 22 = 2 2 , (2.10)
where 2 is the quadrupole deformation parameter of the nucleus, while the parameter describes the degree of triaxiality of the nuclear system, measuring the deviation from axial symmetry. From the above definitions we can rewrite R(✓, ) as

R(✓, ) = R 0 n 1 + 2 r 5 16⇡ [cos (3cos 2 ✓ 1) + p 3sin sin 2 ✓cos2 ] o . (2.11)
The increments of the three axes in the body-fixed frame can be written in terms of the 2 and parameters as follow

R k R 0 R 0 = 2 r 5 4⇡ cos( 2⇡k 3 ), k = 1, 2, 3, (2.12)
where k = 1, 2, 3 correspond to the x, y, and z direction, respectively.

Figure 2.3: Plot of the Eq. 2.12, for k=1, 2, 3, corresponding to the increase in the axis lengths in the x, y, and z directions. Figure adopted from Ref. [START_REF] Greiner | Nuclear models[END_REF].

To easily see the evolution of the axis lengths with , a plot of the functions

2 cos( 2⇡k
3 ) with various and k are given in Fig. 2.3. One can see that at = 0 the nucleus is elongated along the z axis, while the length of x and y axes are equal (axially symmetric). With increasing , the x axis grows at the expense of the y and z axes, until the length of x and z axes are equal at = 60 . This patten is repeated each 60 .

According to the Lund convention [START_REF] Andersson | Nuclear shell structure at very high angular momentum[END_REF], the different nuclear shapes corresponding to the various ( 2 , ) are shown in Fig. 2.4. In this convention, for = 0 and 120 , the nucleus is prolate, while it is oblate for = 60 and 60 . Note that there are two different types of rotations for pure prolate and oblate shapes which can be collective-and non-collective. Collective rotation, which occurs when = 0 or = 60 , the rotational axis of nucleus is perpendicular to the symmetry axis. For non-collective rotation which occurs when = 60 or = 120 , the rotational axis of nucleus is along the symmetry axis. When is not a multiple of 60 one has a triaxial shape. There are discrete symmetries, namely, one can interchange all three axes without changing the shape, which mean an invariance under the point group D 2 [START_REF] Ring | The nuclear many-body problem[END_REF]. However, the interval 0 < < 60 is sufficient to describe all the = 2 shapes.

The Nilsson model

The deformed shell model was originally introduced by S. G. Nilsson [START_REF] Gösta | Binding states of individual nucleons in strongly deformed nuclei[END_REF], which is also often referred to the Nilsson model. In this model, the anisotropic harmonic oscillator potential is used as average field, so the single-particle Hamiltonian is expressed as,

H = h2 2m r 2 + m 2 (! 2 x x 2 + ! 2 y y 2 + ! 2 z z 2 ) C l • s Dl 2 , (2.13)
where ! x , ! y , and ! z are the oscillator frequencies, x, y, and z are the coordinates of a particle in the intrinsic reference system. The parameters C and D control the strength of the spin-orbit and l 2 term, respectively. However, for large N quantum numbers, the l 2 -term shift is too strong and Nilsson replaced the last term with the following form

D(l 2 < l 2 > N ), (2.14)
where < l 2 > N = 1 2 N (N + 3) is the expectation values of l 2 averaged over one major shell with quantum number N. The three frequencies are chosen proportional to the inverse of the half axes a x , a y , and a z of the ellipsoid:

! i = ! o R 0 a i i = x, y, z. (2.15)
If one only take into account the anisotropic harmonic oscillator Hamiltonian, the eigenstates are characterized by the number of oscillator quanta n x , n y , n z , and the eigenvalues are

✏ 0 (n x , n y , n z ) = h! x (n x + 1 2 ) + h! y (n y + 1 2 ) + h! z (n z + 1 2
).

(2.16)

In the case of axially symmetric shapes, one usually choose the z-axis as symmetry axis and further introduce one single parameter of deformation , so

! 2 ? = ! 2 x = ! 2 y = ! 2 ( ) ✓ 1 + 2 3 ◆ , ! 2 z = ! 2 ( ) ✓ 1 4 3
◆ .

(2.17)

The condition of constant volume of the nucleus leads to

! x ! y ! z = const. = ! 3 0 , (2.18)
! 0 is the value of !( ) for = 0. From Eq. 2.17 and Eq. 2.18, we can get

!( ) = ! 0 (1 4 3 2 16 27 3 ) 1/6 , (2.19)
leading in second order to

!( ) = ! 0 [1 + ( 2 3 ) 2 ]. (2.20)
The quantity is related to the deformation parameter 2 , as follows [47]

⇡ 3 2 r 5 4⇡ 2 ⇡ 0.95 2 .
(2.21)

In the case of axial symmetry, it is more convenient to use cylindrical coordinates, and in this case the eigenvalues can be written as Now, if one treats Eq. 2.13 by using the first order perturbation theory [START_REF] Ragnarsson | Shapes and shells in nuclear structure[END_REF], we can obtain

✏ 0 (n z , n ⇢ , n l ) = h! z (n z + 1 2 ) + h! ? (2n ⇢ + m l + 1) ⇡ h! 0 (N + 3 2 ) + ( N 3 n z ), (2.22
E =hNn z ⇤⌦|H|Nn z ⇤⌦i =(N + 3 2 )h! 0 + 1 3 h! 0 (N 3n z ) 2h⇤⌃ µ 0 h! 0 (⇤ 2 + 2n ? n z + 2n z + n ? N (N + 3) 2 ), (2.24)
where N is the principal quantum number, ⌃ is projection of the spin of the particle s on the symmetry axis, ⇤ is projection of the orbital angular momentum on the symmetry axis, and n z is number of oscillator quanta (see Fig. 2.5).

The Nilsson orbitals are often denoted by a set of quantum numbers

⌦ ⇡ [Nn z ⇤],
where ⌦ is projection of single-particle angular momentum on the symmetry axis, and ⇡ is the parity of the state [⇡ = ( 1) l = ( 1) N ].

Figure 2.5: Schematic of the quantum numbers which can describe the deformed nucleus. ⇤, ⌦, ⌃, and K are the projections of the orbital angular momentum l, of the total angular momentum of the particle j, of the spin of the particle s, and of the total angular momentum J onto the symmetry axis, respectively. In addition, R is the angular momentum of the core and M is the projection of the total angular momentum onto the laboratory axis.

Particle-rotor model

The unified nuclear model and its consequences, especially for the nuclear properties pertaining to the ground states and to the low energy region of excitation has been proposed by Bohr and Mottelson in 1953 [50]. It core describes the interplay between the motion of particles and of the collective. In this model, one considers only a few valence particles which move more or less independently in the deformed potential well of the core, and one couple them to a collective rotor which represents the rest of the particles.

The simplest model consists of a particle in a single shell coupled to a rotor. Generally speaking, one divides the Hamiltonian into two parts: an intrinsic part H intr , which describes microscopically a valence particle, and a collective rotor part H rot , which describes the rotations of the inert core. The total Hamiltonian is written as

H = H intr + H rot .
(2.25)

The intrinsic part has the form

H intr = 4 X i=1 X ⌫ " i,⌫ a † i,⌫ a i,⌫ + 1 4 X klmn vklmn a † k a † l a n a m , (2.26)
where " i,⌫ is the single-particle energy in the i-th single-j shell and v is the interaction between the valence particles (residual interaction) which is often neglected.

The collective rotor Hamiltonian reads:

H rot = R 2 1 2J 1 + R 2 2 2J 2 + R 2 3 2J 3 , (2.27)
where the R i are the body-fixed components of the collective angular momentum of the core. The total angular momentum is I = R + j, where j is the angular momentum of the valence nucleons. Thus, H rot can be expanded into three parts:

H rot = 3 X i=1 I 2 i 2J i + 3 X i=1 j 2 i 2J i 3 X i=1 I i j i J i , i = 1, 2, 3. (2.28)
The first term is a pure rotational operator of the rotor which acts only on the intrinsic coordinates; the second term called recoil term, acts on the coordinates of the valence particle; the third term couples the degrees of freedom of the valence particles to the degrees of the freedom of the rotor.

According to the physical situation, there are two important limit cases of the above Hamiltonian, namely, strong coupling (deformation alignment) and decoupling (rotation alignment).

Strong coupling

Assuming that the rotor has the 3-axis being the symmetry axis, the moments of inertia J 1 = J 2 = J and K = ⌦, the corresponding rotor Hamiltonian can be rewritten as

H rot = 1 2J ⇥ (I 1 j 1 ) 2 + (I 2 j 2 ) 2 ⇤ = 1 2J ⇥ I 2 + (j 2 1 + j 2 2 j 2
3 ) (I

+ j + I j + ) ⇤ , (2.29)
where

I ± = I 1 ± I 2 and j ± = j 1 ± j 2 .
The term (I + j + I j + ) corresponds to the classically Coriolis and centrifugal forces, which generates a coupling between the particle motion and the collective rotation. For small I, we can assume that this term is small and we just need to take into account its diagonal contributions, i.e., it's treated in first order perturbation theory. This approximation, in which it is assumed that the influence of the rotational motion in the intrinsic frame of the nucleus can be ignored, is always referred to as the adiabatic approximation or the strong coupling limit [START_REF] Ragnarsson | Shapes and shells in nuclear structure[END_REF].

In this strong coupling case, K is a good quantum number. The angular momentum j of the valence particle is strongly coupled to the the motion of the rotor (see Fig. 2.8), leading to j perpendicular to the R: this gives rise to a band I = K, I = K + 1, I = K + 2, .... . The total energy is given by

E IK = E K + 1 2J ⇥ I(I + 1) K 2 ) ⇤ , K 6 = 1 2 , (2.30)
where E K is the quasiparticle energy.

For K = 1 2 , the total energy is

E IK = E K + 1 2J ⇥ I(I + 1) + a( 1) I+1/2 (I + 1/2) ⇤ , (2.31)
where a is the decoupling factor which is calculated by

a = X nj |C nj | 2 ( 1) j+1/2 (j + 1 2
).

(2.32)

The decoupling limit

In the decoupling case, the Coriolis force is strong and the coupling of the active particle to the deformed core can be neglected. The total angular momentum I is parallel to the single-particle angular momentum j (see Fig. 2.9). This gives rise to the typical I = 2 bands. For a more detailed discussion see Refs. [START_REF] Ring | The nuclear many-body problem[END_REF][START_REF] Ragnarsson | Shapes and shells in nuclear structure[END_REF].

Four-j shells particle-rotor model

Theoretically, various particle-rotor models have been developed to investigate the exotic nuclear structure, in particular the nuclear chirality. For example, one-particle-one-hole PRM combined with the tilted axis cranking (TAC) approximation was first developed to study the chirality in the oddodd nuclei [START_REF] Frauendorf | Tilted rotation of triaxial nuclei[END_REF]. Later, in order to treat more than one valence proton and one valence neutron, and also to study the nuclear chirality, the n-particle-n-hole PRM with nucleons in two single-j shells [51,[START_REF] Qi | Chirality in odd-a rh isotopes within the triaxial particle rotor model[END_REF] and three single-j shells [START_REF] Ayangeakaa | Evidence for multiple chiral doublet bands in 133 Ce[END_REF][START_REF] Kuti | Multiple chiral doublet bands of identical configuration in 103 Rh[END_REF][START_REF] Lieder | Resolution of chiral conundrum in 106 Ag: Dopplershift lifetime investigation[END_REF][START_REF] Qi | Possible multiple chiral doublet bands in 107 Ag[END_REF] have been developed, respectively. In the present work, we observed five pairs of doublet bands in 136 Nd, with configurations involving four different single-j shells, which can not be treated by any PRM model available at the time when the experimental results were obtained. Inspired by the new obtained results in 136 Nd, a n-particle-n-hole version of the PRM with nucleons in four single-j shells was developed [55].

The total Hamiltonian of the PRM model which couples nucleons in four single-j shells to a triaxial rotor is similar to that of Eq. 2.25. The collective rotor Hamiltonian H rot is expressed as

H rot = 3 X k=1 R 2 k 2J k = 3 X k=1 (I k j k ) 2 2J k , (2.33)
where the indexes k = 1, 2, and 3 refer to the three principal axes of the body-fixed frame, R k , I k , and j k denote the angular momentum of the core, of the total nucleus, and of the valence nucleon, respectively. The moments of inertia of the irrotational flow [START_REF] Ring | The nuclear many-body problem[END_REF] are adopted, i.e., J k = J 0 sin 2 ( 2k⇡/3). Additionally, the intrinsic Hamiltonian for valence nucleons is written as

H intr = 4 X i=1 X ⌫ " i,⌫ a † i,⌫ a i,⌫ , (2.34)
where " i,⌫ is the single-particle energy in the i-th single-j shell given by

h sp = ± 1 2 C n cos ⇥ j 2 3 j(j + 1) 3
⇤ + sin 2 p 3 j 2 + + j 2 o , (2.35)
where the plus sign refers to a particle, the minus to a hole, and the parameter C is responsible for the level splitting in the deformed field and directly proportional to the quadrupole deformation 2 as in Ref. [START_REF] Shou-Yu | Coupling parameter in the single-j shell model[END_REF]. The single-particle state and its time reversal state are expressed as

a † ⌫ |0i = X ↵⌦ c ⌫ ↵⌦ |↵, j⌦i, (2.36)
a † ⌫ |0i = X ↵⌦ ( 1) j ⌦ c ⌫ ↵⌦ |↵, j ⌦i, (2.37)
where ⌦ is the projection of the single-particle angular momentum j along the 3-axis of the intrinsic frame and restricted to . . . , 3/2, 1/2, 5/2, . . . due to the time-reversal degeneracy, and ↵ denotes the other quantum numbers.

For a system with P 4 i=1 N i valence nucleons (N i denotes the number of the nucleons in the i-th single-j shell), the intrinsic wave function is given as

|'i = 4 Y i=1 ⇣ n i Y l=1 a † i,⌫ l ⌘⇣ n 0 i Y l=1 a † i,μ l ⌘ |0i, (2.38)
with n i + n 0 i = N i and 0  n i  N i . The total wave function can be expanded into the strong coupling basis

|IMi = X K' c K' |IMK'i, (2.39)
with

|IMK'i = 1 p 2(1 + K0 ', ') ⇥ |IMKi|'i + ( 1) I K |IM Ki| 'i ⇤ , (2.40)
where |IMKi is the Wigner function ) being an even integer. Once obtained the wave functions of the PRM, the reduced transition probabilities B(M 1) and B(E2) can be calculated with the M 1 and E2 operators [START_REF] Zhang | Chiral bands for a quasiproton and quasi-neutron coupled with a triaxial rotor[END_REF].

q 2I+1 8⇡ 2 D I MK ,
Note that due to the inclusion of n-particle-n-hole configurations with four single-j shells, the size of the basis space is quite large. It is rather timeconsuming in the diagonalization of the PRM Hamiltonian matrix. In order to solve this problem, a properly truncated basis space by introducing a cutoff for the configuration energy P i,⌫ " i,⌫ was adapted which is similar in the shellmodel-like approach (SLAP) [START_REF] Zeng | Particle-number-conserving method for treating the nuclear pairing correlation[END_REF][START_REF] Shi | Shell-modellike approach based on cranking covariant density functional theory: Band crossing and shape evolution in 60 Fe[END_REF]. In this way, the dimension of the PRM matrix is reduced to ⇠ 5000-10000, , while maintaining the energy uncertainty within 0.1% [55].

This new version of the PRM has been applied to investigate the energy spectra, the electromagnetic transition probabilities, as well as the angular momentum geometries of the five pairs of nearly degenerate doublet bands in 136 Nd, which are discussed in detail in Section. 5.2.2.

Cranked Nilsson-Strutinsky model

One of the most important and widely used models in the high-spin study is the cranked Nilsson-Strustinsky (CNS) model [START_REF] Afanasjev | Termination of rotational bands: disappearance of quantum many-body collectivity[END_REF][START_REF] Bengtsson | Rotational bands and particle-hole excitations at very high spin[END_REF][START_REF] Carlsson | Calculating the nuclear mass in the very high angular momentum regime[END_REF][START_REF] Afanasjev | Gradual loss of collectivity in rotational bands in the A ⇠ 110 (Z 50, N ⇠ 60) mass region[END_REF], which successfully describes the high-spin states of nuclei. In the present work, this model was applied to discuss the various band structures in 136 Nd (see Section. 5.2.3). In the Nilsson-Strustinsky method, the total energy of the nucleus is split into an average part, parametrized by a macrosopic expression, and a fluctuating part extracted from the variation of the level density around the Femi surface. The microscopic part using the Strustinsky shell correction [START_REF] Afanasjev | Gradual loss of collectivity in rotational bands in the A ⇠ 110 (Z 50, N ⇠ 60) mass region[END_REF].

Cranking model

To get a deeper insight into the properties of rotating nuclei from a microscopic point of view, the cranking model was intruduced by Inglis [START_REF] Inglis | Particle derivation of nuclear rotation properties associated with a surface wave[END_REF][START_REF] Inglis | Nuclear moments of inertia due to nucleon motion in a rotating well[END_REF]. The 2.5. Cranked Nilsson-Strutinsky model basic idea of the cranking model is the following classical assumption: if one introduces a coordinate system which rotates with constant angular velocity ! around x-axis, the motion of the nucleon in the rotating frame is rather simple when the angular frequency is properly chosen; in particular, the nucleons can be thought of as independent particles moving in an average potential well which is rotating with the coordinate frame [START_REF] Ring | The nuclear many-body problem[END_REF]. The single-particle Hamiltonian in the rotating system h ! is given as,

h ! = h !j x , (2.41)
where h is the single-particle Hamiltonian in the laboratory system and j x is the x-component of the single-particle angular momentum. Thus, the total energy in the laboratory system is calculated as,

E tot = X i < ! i |h| ! i >= X i e ! i + X i ! < ! i |j x | ! i >, (2.42)
and the total spin I is

I ⇡ I x = X i < ! i |j x | ! i >, I 1, (2.43)
where ! i are the single-particle eigenfunction in the rotating system and e ! i =< ! i |h w | ! i > the corresponding eigenvalues, also called single-particle Routhians. The slope of the Routhian corresponds to the alignments i x . It is written as

i x = de ! i d! .
(2.44)

Note that the Eq. 2.41 Hamiltonian is dependent on the rotational frequency !, and therefore it breaks the time-reversal symmetry, but remains invariant with respect to space inversion (parity invariance), which means that the parity (⇡) is a good quantum number. In addition, the cranking Hamiltonian is invariant with respect to rotation through an angle ⇡ around the rotating axis [START_REF] Afanasjev | Termination of rotational bands: disappearance of quantum many-body collectivity[END_REF].

R x (⇡) = e i⇡jx , (2.45)
with eigenvalue

r = e i⇡↵ , (2.46)
where ↵ is the signature quantum number. The single-particle orbitals can be classified with respect to the signature quantum number ↵, which can take values

↵ i = + 1 2 (r i = i) or ↵ i = 1 2 (r i = i).
In addition, the signature ↵ relates to the total angular momentum. For a system with an even number of nucleons, ↵ = 0 (r = +1), I = 0, 2, 4... , ↵ = 1 (r = 1), I = 1, 3, 5... .

(2.47)

For a system with an odd number of nucleons,

↵ = + 1 2 (r = i), I = 1 2 , 5 2 , 9 2 ... , ↵ = 1 2 (r = +i), I = 3 2 , 7 2 , 11 2 ... . (2.48)
As mentioned above, the parity ⇡ is a good quantum number, which together with the signature ↵ can be used to describe the Routhians of the quasiparticles.

The advantage of using the cranking model is that it gives a microscopic description of rotating nuclei, in which the total angular momentum is the sum of single-particle angular momenta, and thus collective and non-collective rotations can be treated on the same footing. However, this model is a semiclassical approximation due to the fact that the rotation is imposed externally. A fixed rotation axis is used in the model which also breaks the rotational invariance. In addition, another important shortcoming of the cranking model is that the wave functions are not eigenstates of the angular momentum operator, which causes difficulties, i.e., a proper calculation of the electromagnetic transition probabilities [START_REF] Ragnarsson | Shapes and shells in nuclear structure[END_REF][START_REF] Afanasjev | Termination of rotational bands: disappearance of quantum many-body collectivity[END_REF].

The rotating harmonic oscillator

Many features of the high-spin structure of nuclei can be easily illustrated with the rotating (cranked) harmonic oscillator potential because of its simplicity, and allowing to express in analytic form all matrix elements.

If only the orbital angular momentum is considered, the cranking Hamiltonian reads

h ! = h osc !l 1 , (2.49)
where

h osc = h2 2m 4 + m 2 (! 2 1 x 2 1 + ! 2 2 x 2 2 + ! 2 3 x 2 3 ), (2.50)
and

l 1 = x 2 p 3 x 3 p 2 .
(2.51)

The oscillator frequencies ! 1 , ! 2 and ! 3 are expressed in the standard way through the deformation coordinates ✏ 2 and [START_REF] S E Larsson | The nuclear potential-energy surface with inclusion of axial asymmetry[END_REF]:

! i = ! 0 (✏ 2 , )[1 2 3 ✏ 2 cos( + i 2⇡k 3 )] i = 1, 2, 3. (2.52)
Introducing boson creation and annihilation operators, a + i and a i , see Ref. [START_REF] Afanasjev | Termination of rotational bands: disappearance of quantum many-body collectivity[END_REF], the cranking Hamiltonian Eq. 2.49 can be written in the form

h w = 3 X i=1 h! i (a + i a i + 1 2 ) !l 1 , (2.53)
where n 1 , n ↵ and n specify the number of quanta in the three normal-mode degrees of freedom (directions).

A further quantity of interest is the expectation value of l 1 . The diagonal parts of this operator are easily obtained and thus, for an orbital that is characterized by the occupation numbers n 1 , n ↵ and n :

< l 1 > ⇡ ( p 1 + p 2 ) 1/2 (n n ↵ ). (2.56)
The total quantities of the A-particle system will now be considered with the A-particles generally filling the lowest or close-to-lowest energy orbitals of the cranked harmonic oscillator. For this purpose, we define the quantities

⌃ k = X ⌫ occ < ⌫|a + k a k + 1 2 |⌫ >= X ⌫ occ (n k + 1 2 ) ⌫ . (2.57)
The index k takes the values k = 1, ↵, and (or k = 1, 2 and 3 for ! = 0) and the summation extends over the occupied orbitals, |⌫ >. Then the specific configuration in the rotating frame is defined by (⌃ 1 , ⌃ ↵ , ⌃ ). The total energy in the rotating system is given by

E = X ⌫ occ h⌫|h w |⌫i = h! 1 ⌃ 1 + h! ↵ ⌃ ↵ + h! ⌃ . (2.58)
The total energy in the laboratory system is calculated as the sum of expectation values of the Hamiltonian

h osc E = X ⌫ occ h⌫|h osc |⌫i = X ⌫ occ h⌫|h ! + !l 1 |⌫i = E ! + !I. (2.59)
In addition, for a fixed configuration (⌃ 1 , ⌃ ↵ , ⌃ ) and spin I, the minimized energy E can be obtained.

The cranked single-particle Nillsson Hamiltonian

The cranked single-particle Nilsson Hamiltonian h ! is given by [START_REF] Bengtsson | Rotational bands and particle-hole excitations at very high spin[END_REF]

h ! = h h.o. (✏ 2 , ) + 2h! 0 ⇢ 2 ✏ 4 V 4 ( ) + V 0 !j x , (2.60)
where h h.o. (✏ 2 , ) is the anisotropic harmonic oscillator Hamiltonian, ⇢ is the radius in the stretched coordinate system. The hexadecapole deformation potential V 4 ( ) is defined to obtain a smooth variation in the -plane, so it does not break the axial symmetry when = 120 , 60 , 0 , and 60 . It has the form

V 4 ( ) = a 40 Y 40 + a 42 (Y 42 + Y 4 2 ) + a 44 (Y 44 + Y 4 4 ), (2.61)
where the parameters a 4i are chosen as

a 40 = 1 6 (5cos 2 + 1), a 42 = 1 12 p 30sin2 , a 44 = 1 12 p 70sin 2 . (2.62)
The term V 0 is the Nilsson potential which as defined in Ref. [START_REF] Gösta | Binding states of individual nucleons in strongly deformed nuclei[END_REF].

The diagonalization of the Hamiltonian of Eq. 2.60 gives the eigenvalues e ! i and eigenfunctions of the eigenvectors ! i . Furthermore, the single-particle energies in the laboratory system and the single-particle angular momentum alignments in the rotating system can be obtained.

Note that the sum over the occupied single-particle states from phenomenological potentials, such as the Nilsson or Woods-Saxon potentials, turned out to yield poor approximations to average nuclear properties and their deformation dependence [START_REF] Ring | The nuclear many-body problem[END_REF]. To overcome these problems, the total energies are renormalized to a rotating liquid drop behaviour.

The rotating liquid drop model

The energy of a rotating nuclear liquid drop at the fixed spin I 0 has the form

E RLD (I 0 ) = E LD (I = 0) + 1 2J rig (✏) I 2 0 , (2.63)
where J rig (✏) is the rigid body moment of inertia and ✏ =(✏ 2 , , ✏ 4 ...). The total energy in the liquid-drop model is given by [START_REF] Afanasjev | Termination of rotational bands: disappearance of quantum many-body collectivity[END_REF]

E RLD (I 0 ) = a v ⇥ 1 k v ( N Z A ) 2 ⇤ A + 3 5 e 2 Z 2 R c [B c (✏) 5⇡ 2 6 d R c ) 2 ] + a s [1 k s ( N Z A ) 2 ]A 2/3 B s (✏), (2.64)
where B c (✏) and B s (✏) are the coulomb and surface energies of a nucleus with a sharp surface in units of their corresponding values at a spherical shape [START_REF] Afanasjev | Termination of rotational bands: disappearance of quantum many-body collectivity[END_REF], respectively. The second term in the Coulomb energy is a (shape-independent) diffuseness correction with d being the diffuseness. In addition, we often define the Coulomb energy constant a c as a c = 3

The configuration-dependent CNS approach

Following the standard Nilsson-Strutinsky method [START_REF] Bengtsson | Rotational bands and particle-hole excitations at very high spin[END_REF][START_REF] Strutinsky | Shell effects in nuclear masses and deformation energies[END_REF], the total nuclear energy E tot at a specific deformation ✏ and a specific spin I 0 is obtained as a sum of the rotating liquid drop energy and the shell correction energy

E tot (✏, I 0 ) = E LD (✏, I = 0) + 1 2J rig (✏) I 2 0 + E sh (✏, I 0 ). (2.65)
The shell correction energy is defined as the difference between the discrete and smoothed single-particle energy sums,

E sh (I 0 ) = X e i (!, ✏) I=I 0 X e e i (e !, ✏) e I=I 0 , (2.66)
where the smoothed sum (indicated by ⇠) is calculated using the Strutinsky procedure [START_REF] Ragnarsson | Shapes and shells in nuclear structure[END_REF][START_REF] Strutinsky | Shell effects in nuclear masses and deformation energies[END_REF]. In the numerical calculations, it is parametrized as

X e e i (e !, ✏) e I=I 0 = X e e i (e !, ✏) e I=0 + 1 2J str (✏) I 2 0 + bI 4 0 , (2.67)
where J str is the (Strutinsky) smoothed moment of inertia and E 0 is the value of P e e i (e !, ✏)| e I=I 0 for e I = 0. The E 0 , I 2 0 , and bI 4 0 terms were introduced in Ref. [START_REF] Andersson | Nuclear shell structure at very high angular momentum[END_REF]. The constants J str , E 0 , and b are determined by calculating the smoothed sum at the different frequencies (see Ref. [START_REF] Andersson | Nuclear shell structure at very high angular momentum[END_REF]). Using these formulas, we can calculate the total nuclear energy as a function of spin I at any deformation.

Furthermore, in oder to get a better physical understanding of the relation between the level density and the shell energy, it is useful to rewrite these equations slightly and make some approximations. A quasi-shell energy is defined as

E quasi sh (!, ✏) = X e i (!, ✏) X e e i (!, ✏). (2.68)
This definition is important because it shows that the numerical values of E sh and E quasi sh are very similar. The E sh is defined exactly analogous to the static shell energy. Thus, ! enters very much as a deformation and we can take over all our experience from the static case.

In order to obtain the renormalized frequency, the formulas above are combined by rewriting the total energy as

E tot (I 0 ) = X e i (!, ✏) I=I 0 + E LD (✏, I = 0) E 0 + n 1 2J rig 1 2J str o I 2 0 bI 4 0 . (2.69)
With the frequency given by @E/@I, the resulting renormalized frequency is

! ren (I 0 ) = ! + ⇣ 1 2J rig 1 2J str ⌘ I 0 + bI 4 0 , (2.70)
where b value is always very small, thus we can neglect the bI 4 0 term. Furthermore, for highly deformed bands with no pairing I = J str !, using this approximation, we can obtain

! ren = (J str /J rig )!.
(2.71)

With this renormalization being in the range 1.2-1.3, depending on the deformation, frequencies in Routhian diagrams are directly comparable with experimentally observed frequencies only if the Routhians are plotted as a function of ! ren . Note that in the present used version of the cranked Nilsson-Strutinsky model the pairing correlations have been neglected. The neglect of pairing has an advantage in that the tracing of a fixed configuration undergoing drastic deformation changes becomes possible over the considerable spin range of a band up to termination. Furthermore, in the present study, we are mainly interested in the very high spin states of nuclei, for which the deformation changes play an more important role than the pairing.

Tilted axis cranking covariant density functional theory

Magnetic rotation is an exotic rotational phenomenon observed in weakly deformed or near-spherical nuclei which are interpreted in terms of the shears mechanism [START_REF] Macchiavelli | The competition between the shears mechanism and core rotation in a classical particlesplus-rotor model[END_REF]. Since their first observation, the magnetic bands have been mainly investigated in the framework of tilted axis cranking (TAC) [START_REF] Frauendorf | Tilted rotation of triaxial nuclei[END_REF][START_REF] Frauendorf | Tilted cranking[END_REF][START_REF] Frauendorf | Description of multi-quasiparticle bands by the tilted axis cranking model[END_REF]. In the last decades, the covariant density functional theory (CDFT) and its extensions have been proved to be successful in describing a series of nuclear properties, like ground-states and excited states, radii, single-particle spectra, magnetic rotation, and collective motions, etc. Recently, the tilted axis cranking covariant density functional theory (TAC-CDFT) [START_REF] Meng | Progress on tilted axis cranking covariant density functional theory for nuclear magnetic and antimagnetic rotation[END_REF][START_REF] Vretenar | Relativistic hartree-bogoliubov theory: static and dynamic aspects of exotic nuclear structure[END_REF][START_REF] Meng | Relativistic continuum hartree bogoliubov theory for ground-state properties of exotic nuclei[END_REF][START_REF] Sulaksono | The nonrelativistic limit of the relativistic point coupling model[END_REF][START_REF] Zhao | Novel structure for magnetic rotation bands in 60 Ni[END_REF] has been developed and applied for the description of the chiral bands.

Tilted axis cranking

In the standard cranking model, it is assumed that the rotational axis coincides with one of the principal axes of the deformed density distribution. Such principal axis cranking (PAC) solutions always exist. In the TAC approach, the rotational axis does not coincide with any principal axis. At difference from the PAC model, the signature is not a good quantum number. The TAC model is based on the mean field theory that permits the calculation of the orientation of the deformed field in space together with the parameters that define its shape. Thus, one may easily study multi-quasiparticle excitations, and the consequences of changes of the deformation or the pairing. It 2.6. Tilted axis cranking covariant density functional theory gives transparent classical vector diagram pictures of the angular momentum coupling, which is of great help to understand the structure of rotation bands. The drawback of the TAC model is that it cannot describe the gradual onset of signature splitting as well as the mixing of bands with substantially different quasiparticle angular momentum as in the standard cranking theory [START_REF] Meng | Progress on tilted axis cranking covariant density functional theory for nuclear magnetic and antimagnetic rotation[END_REF].

Assuming that the rotational axis is the z-axis and with a constant angular velocity !, the corresponding Hamiltonian in intrinsic frame is given as

H 0 = H !J z . (2.72)
In Ref. [START_REF] Frauendorf | Description of multi-quasiparticle bands by the tilted axis cranking model[END_REF], the H term is defined as a sum of the spherical Hamiltonian and the pairing plus quadrupole interaction,

H = H sph 2 2 X µ= 2 Q + µ Q µ GP + P N, (2.73)
with the spherical part

H sph = X k ✏ k c + k c k , (2.74)
where H sph is parameterized in the same way as the Nilsson Hamiltonian, the pairing interaction is defined by the monopole pair field P + , the quadrupole interaction is defined by the operators Q µ , and is the chemical potential.

As written above all expressions are only accounting for one kind of particles. Thus, they are understood as sums of a proton and a neutron part, and the terms GP + P and N for both protons and neutrons. In practice, the actual values of the force constants and G depend on the configuration space under consideration and are determined from experimental data. A detailed description of the TAC model can be found in the Refs. [START_REF] Ring | The nuclear many-body problem[END_REF][START_REF] Frauendorf | Spontaneous symmetry breaking in rotating nuclei[END_REF][START_REF] Frauendorf | Tilted cranking[END_REF][START_REF] Frauendorf | Description of multi-quasiparticle bands by the tilted axis cranking model[END_REF]. As in the present work we are interested in the chiral motion, in the following we will examine the TAC solutions for specific rotation axes.

Rotation about a principal axis

If the axis of rotation (z) coincides with one of the principal axes, the orientation angles defined as in the case of spherical coordinates, are satisfying the condition:

✓ = 0, ⇡ 2 ; = 0, ⇡ 2 . (2.75)
The upper panel of Fig. 2.10 shows this case, in which J has the direction of one of the principal axes and R z (⇡) satisfies the relation

R z (⇡)|i = e i↵⇡ |i, (2.76)
which means that the signature quantum number ↵ takes the values I = ↵+2n, n = ±1, ±2..., giving rise to a band characterized by I = 2 transitions.

Rotation around an axis in a principal plane

If the axis of rotation (z) does not coincide with any of the principal axes, but still lies in one of the three principal planes, we call such solution planar. The orientation angles satisfy the relations:

✓ 6 = 0, ⇡ 2 ; = 0, ⇡ 2 , or ✓ = 0, ⇡ 2 ; 6 = 0, ⇡ 2 .
(2.77)

The middle panel of Fig. 2.10 displays this case, in which the axis of rotation lies in one of the three principal planes 1-2, 2-3 or 1-3 and the R z (⇡) symmetry is broken, that is it satisfies the relation

R z (⇡)|i 6 = e i↵⇡ |i, (2.78)
which means the signature is not a good quantum number anymore and thus there is no restriction on the angular momentum I, giving rise to a band characterized by I = 1 transitions..

Rotation around an axis out of a principal plane

If the axis of rotation (z) is out of the three principal planes, we call such a solution aplanar, and the orientation angles satisfy the relations

✓ 6 = 0, ⇡ 2 ; 6 = 0, ⇡ 2 .
(2.79)

The lower panel of Fig. 2.10 displays the aplanar case in which the rotational axis does not lie in one of the principal planes. In this case, T R y (⇡) symmetry is broken. The T R y (⇡) is the time-reversal operator combined with rotation about the y-axis by ⇡. The two combinations have opposite chirality, corresponding the left-and right-handed frames. The breaking of the T R y (⇡) symmetry causes two degenerate doublet I = 1 bands with the same parity.

TAC-CDFT

The covariant density functional theory (CDFT) can be traced back to the successful relativistic mean-field (RMF) models introduced by Walecka and Serot [START_REF] Serot | The relativistic nuclear many body problem[END_REF][START_REF] Brian | Recent progress in quantum hadrodynamics[END_REF], which was further developed by many physicists. In the following, we will give a brief description of the tilted axis cranking covariant density functional theory (TAC-CDFT) based on the point-coupling interaction [START_REF] Zhao | Novel structure for magnetic rotation bands in 60 Ni[END_REF].

The starting point of the CDFT is an effective Lagrangian density of the form

L = L free + L 4f + L hot + L der + L em , (2.80)
which is divided into five terms. The Langrangian density for free nucleons

L free , L free = ¯ (i µ @ m) , (2.81)
the four fermion point-coupling terms L 4f ,

L 4f = 1 2 ↵ S ( ¯ )( ¯ ) 1 2 ↵ V ( ¯ µ )( ¯ µ ) 1 2 ↵ T S ( ¯ ⌧ )( ¯ ) 1 2 ↵ T V ( ¯ ⌧ µ )( ¯ ⌧ µ ), (2.82)
the higher order terms L hot accounting for the medium effects,

L hot = 1 3 S ( ¯ ) 3 1 4 S ( ¯ ) 4 1 4 V [( ¯ µ )( ¯ µ )] 2 , (2.83)
the derivative terms L der to simulate the effects of finite-range,

L der = 1 2 S @ ⌫ ( ¯ )@ ⌫ ( ¯ ) 1 2 V @ ⌫ ( ¯ µ )@ ⌫ ( ¯ µ ) 1 2 T S @ ⌫ ( ¯ ⌧ )@ ⌫ ( ¯ ) 1 2 T V @ ⌫ ( ¯ ⌧ µ ), (2.84)
Chapter 2. Theoretical backgroud 51 and the electromagnetic interaction term L em ,

L em = 1 4 F µ⌫ F µ⌫ e 1 ⌧ 3 2 ¯ µ A µ , (2.85)
where m is the nucleon mass and e is the charge unit for protons. A µ and F µ⌫ are the four-vector potential and field strength tensor of the electromagnetic field, respectively. There are in total 11 coupling constants, ↵ S , ↵ V , ↵ T S , ↵ T V , S , S , V , S , V , T S , and ↵ T V , in which ↵ refers to the four-fermion term, and respectively to the third-and fourth-order terms, and the derivative couplings. The subscripts S, V , and T respectively indicate the symmetries of the couplings, i.e., S stands for scalar, V for vector, and T for isovector [START_REF] Zhao | New parametrization for the nuclear covariant energy density functional with a point-coupling interaction[END_REF].

The Hamiltonian density, i.e., the 00 components of the energy-momentum tensor, can be obtained by the Legendre transformation

H = T 00 @L @ ˙ i ˙ i L, (2.86)
where i is the nucleon or photon field. Thus, the total Hamiltonian is given as

H = Z d 3 xH. (2.87)
In TAC-CDFT, it's assumed that the nucleus rotates around an axis in the xz plane, the Lagrangian in Eq. 2.80 is transformed into a frame rotating uniformly with a constant rotational frequency,

⌦ = (⌦ x , 0, ⌦ z ) = (⌦cos✓ ⌦ , 0, ⌦sin✓ ⌦ ), (2.88)
where ✓ ⌦ is the tilted angle between the cranking axis and the x-axis. The Dirac equation with nucleon potentials has the form

[↵ • ( ir V ) + (m + S(r)) + V (r) ⌦ • Ĵ] k = ✏ k k , (2.89)
where Ĵ is the total angular momentum of the nucleon spinors, ✏ k represents the single-particle Routhains for nucleons, and the relativistic fields S(r) and V µ (r) are connected in a self-consistent way to the nucleon densities and current distributions, which are obtained from the single-nucleon spinors k .

The iterative solution of these equations yields single-particle energies, expectation values for the three components h Ĵi i of the angular momentum, total energies, quadrupole moments, transition probabilities, etc. The magnitude of the angular velocity ! is connected to the angular momentum quantum number I by the semiclassical relation h Ĵi 2 = I(I + 1), and its orientation is determined by minimizing the total Routhian self-consistently. The detailed formalism of TAC-CDFT can be seen in Refs. [START_REF] Meng | Progress on tilted axis cranking covariant density functional theory for nuclear magnetic and antimagnetic rotation[END_REF][START_REF] Vretenar | Relativistic hartree-bogoliubov theory: static and dynamic aspects of exotic nuclear structure[END_REF][START_REF] Meng | Relativistic continuum hartree bogoliubov theory for ground-state properties of exotic nuclei[END_REF][START_REF] Sulaksono | The nonrelativistic limit of the relativistic point coupling model[END_REF][START_REF] Zhao | Novel structure for magnetic rotation bands in 60 Ni[END_REF][START_REF] Zhao | New parametrization for the nuclear covariant energy density functional with a point-coupling interaction[END_REF].

In the present work, the TAC-CDFT calculations have been applied for the doublet bands built on the 4-and 6-quasiparticle configurations identified in 136 Nd, which will be presented in Section. 5.2.1.

Transition probabilities

Transition probabilities

The ratios of reduced transition probabilities (B(M 1)/B(E2)) are crucial in the intrinsic nuclear structure studies. B(M 1)/B(E2) ratios can be extracted from the experimental transition energies and intensities, together with the multipole mixing ratio (angular distributions/correlations). It's also possible to obtain them from the measurement of the lifetime of nuclear states. Ratios of reduced transition probabilities can be extracted as

B(M 1; I ! I 1) B(E2; I ! I 2) = 0.697 E 5 (M 1; I ! I 1) E 3 (E2; I ! I 2) I (M 1) I (E2) 1 1 + 2 (2.90)
where I (M 1)/I (E2) is ratio of the the intensities of the rays, denotes the mixing ratio ration of the I = 1 transitions. The B(M 1)/B(E2) ratios are expressed in units of (µ N /eb) 2 and E in MeV.

Theoretically, the B(M 1)/B(E2) ratios can be calculated using the semiclassic formalism of Dönau and Frauendorf [START_REF] Dönau | Proceedings of the conference on high angular momentum properties of nuclei[END_REF][START_REF] Dönau | Electromagnetic radiation of rotating nuclei[END_REF].

Chapter 3 Experimental techniques

One of the goals of the nuclear structure physics is to understand the properties of the high-spin states. Investigations on the nuclear structure at high spins require specific experimental techniques. The population of highspin states in nuclei, the methods of measurements, and the experimental facilities used in this work are presented in this chapter.

Heavy-ion fusion-evaporation reactions

There are many types of reactions to produce excited nuclei, i.e., inelastic scattering, Coulomb excitation, heavy-ion fusion-evaporation reaction (a heavy-ion is defined as a nucleus with mass number A > 4). Among them, the fusion-evaporation is one of the most useful in the high-spin physics since it can produce excited nuclei in very high-spin states. To overcome the Coulomb barrier V c form a compound nucleus, the minimum beam energy required is

V c = 1.44 Z b • Z t R b + R t [MeV], (3.1)
where the subscripts b and t denote projectile and target, respectively. R b,t = r 0 A 1/3 b,t is the radius of respective nucleus and Z b,t represent the atomic numbers. After fusion of the projectile and target, the compound nucleus is in a highly excited state, with the excitation energy of

E ex = Q + E cm , (3.2)
where E cm is the kinetic energy of the system in the center of the mass frame and the Q-value is the reaction energy. They are given by

Q = (M b + M t M cn ), E cm = m t m b + m t E lab. . (3.3)
We can also estimate the maximum angular momentum transferred in a fusion evaporation reaction, l max , which can be expressed as

l 2 max = 2µR 2 h2 (E cm V c ), (3.4)
where µ = R b Rt R b +Rt is the reduced mass of the compound nuclear system. In fusion-evaporation reactions, a compound nucleus with very high excitation energy and angular momentum is formed when a heavy-ion beam bombards a target nucleus. If the compound nucleus does not fission, then the compound hot system will cool down by evaporation of particles, such as neutrons, protons, and ↵ particles. Each of the emitted particles carries away an energy of ⇡ 8 MeV and 1 2 h of angular momentum. Finally, the residual nucleus produced after the evaporation of the particles loses the rest of excitation energy and almost all angular momentum via -ray emission, until it reaches the ground state. The dominant part of deexcitations of the residual nucleus proceeds by -ray emission. Therefore, the -ray spectroscopy as an efficient method was used to the study of excited states of nuclei. Generally, in the -ray spectroscopy one uses energy-sensitive HPGe detectors for the measurement of the rays emitted from a residual nucleus, and one analizes the properties of rays, such as the energy, the multipolarity from angular distribution or angular correlation, the electromagnetic character from polarization. A schematic picture showing the formation and decay of the compound nucleus is given in Fig. 3.1.

Interaction mechanisms of the -rays with matter

In order to understand how to extract the information carried by therays, a knowledge of the -ray interactions with matter is important. Only three major types of interaction play a significant role in nuclear spectroscopy measurements: photoelectric absorption, Compton scattering, and pair production. These processes are schematically shown in Fig. 3.2.

Photoelectric absorption

Photoelectric absorption is an interaction in which the incident -ray photon disappears, namely, is fully absorbed. In this case, a photoelectron is emitted from one of the electron shells (E b ) of the absorber atom with a kinetic energy given by the incident photon energy E minus the binding energy of the electron in its original shell, E e = E E b . The process is shown in Fig. 3.2 (a). After the electron has been emitted, a hole is created in the shell, X-rays are emitted in the process of filling the created hole, but it is also possible that the another electron, called Auger electron, is emitted. For typical -ray energies, the photoelectric effect is most likely to occur in the K shell, which has typical binding energies ranging from a few keV for low-Z materials to tens of keV for the materials with high-Z.

Compton scattering

In this process, the incident -ray with energy E is scattered by an electron, creating a recoil electron and a scattered -ray photon deflected through an angle ✓ with respect to its original direction, with the energies of E e and E 0 , respectively. This process is illustrated in Fig. 3.2 (b). The energy of the scattered photon is given by

E 0 = E 1 + ( E m 0 c 2 )(1 cos✓) , (3.5)
where m o c 2 = 0.511 MeV is the rest mass energy of the electron. The kinetic energy of the recoil electron is

E e = E E 0 = E " 1 1 1 + ( E m 0 c 2 )(1 cos✓) # . (3.6)
From the above equations, two extreme cases can be identified: 1. When the scattering angle ✓ = 180 , the incident -ray is backscattered towards its original direction, while the electron recoils along the incidence direction, the recoil electron having the maximum energy. This is the so-called "Compton edge" in a -ray spectrum, which corresponds to an energy

E e | ✓=180 = E 2 E m 0 c 2 1 + 2 E m 0 c 2 ! . (3.7)
2. When the scattering angle ✓ = 0 , E e ⇡ 0.

For most -ray spectroscopy experiments, Compton scattering is the main cause of false events and decrease the P/T ratio of the spectra. To overcome these drawbacks anti-Compton shields were developed, which will be discussed in the section 3.3.

Pair production

The third effect shown in Fig. 3.2 (c) is pair production. The process occurs when an incident -ray within the strong electric field near the atomic nucleus, and results in the creation of an electron-positron pair. Since an energy of 2m 0 c 2 is required to create the pair, a minimum energy of 1.02 MeV is required to make the process energetically possible. After their production, both the electron and positron travel through the detector medium loosing their energy. Once kinetic energy of the positron becomes very low, it will annihilate or combine with a electron in the absorbing medium and emission of two anti-parallel 0.511 MeV -rays. This process is shown in Fig. 3.2 (c).

High-purity Germanium -ray detector

In the 1960s, the Ge(Li) detectors were developed which marked the birth of high resolution -ray spectroscopy. In the 1980s, the more advanced HPGe detectors became available widespread. The HPGe detector has many advantages over the Ge(Li) detector, i.e., HPGe offer the best compromise between energy resolution and efficiency; it also can be stored at room temperature which reduces storage costs and facilitates easier transportation, while Ge(Li) detectors must be continuously maintained at low temperature [START_REF] Eberth | From ge (li) detectors to gamma-ray tracking arrays-50 years of gamma spectroscopy with germanium detectors[END_REF]. In addition, a detector system based on HPGe has good energy resolution, good P/T ratio, high efficiency and rather good timing properties which allow to extract the events of interest from complex spectra. In the 1990s, nuclear physicists built the next generation of -ray arrays, i.e., GAMMASPHERE and EU-ROBALL, with the aim to realize an ultimate 4⇡ array of escape-suppressed Ge detectors. More recently the first phase of Advanced GAmma Tracking Array (AGATA) was complete, as a new generation -ray spectrometer. AGATA is based on the technique of -ray energy tracking in electrically segmented high-purity germanium crystals [START_REF] Akkoyun | Agataadvanced gamma tracking array[END_REF]. A HPGe detector is generally operated as fully depleted detector. Reversebiasing requires that a positive voltage be applied to the p-n junction, as shown in Fig. 3.3. At the p-n junction, the "built-in" electric field and the applied electric field are in the same direction. A thicker depletion region is created when these two fields add. The depletion region is a region of net zero charge, without free electrons and holes. The -ray interact with the germanium crystal within the depleted zone, producing electron-hole pairs which move towards the detector outer contacts. The produced charge induces a voltage pulse signal in a preamplifier. There are two major types of HPGe detectors: planar and coaxial. The planar detector usually used to measure the complex low-energy -ray in the range of 10 keV to a few hundred keV or X-rays. It achieves the best energy resolution due to low capacitance. Coaxial detectors have a large detection efficiency at high -ray energies, and are used to measure in the range of 10 keV to 10 MeV.

The JUROGAM II array

Compton suppression with BGO detectors

The dominating contribution to the background in -ray spectroscopy is come from the incomplete energy deposition of the incoming photon in the detector volume due to the Compton scattering. The Compton scattered -rays will result a continuous background in the -ray spectrum which consequently raises the detection limit for low energies and also decrease P/T ratio. In order to counteract these unwanted effects, we can suppress the Compton scattered -rays by surrounding the HPGe detector with a secondary veto detector, which usually is a scintillator. The majority of the -rays that are scattered out of the HPGe detector are then detected in the surrounding scintillator. Producing a veto signal that is used to eliminate the partially absorbed events from the spectra of the Ge detector. Generally speaking, the secondary detector is expected to have a high detection efficiency and high density to absorbe the escaping photons as much as possible. The bismuth germanate (BGO, Bi 4 Ge 3 O 12 ) material is the preferred choice, because of its excellent timing properties, high Z, high density (7.3 g/cm 3 ), and light output as compared with NaI(Tl). If Compton scattered -rays escape from the HPGe detector and are detected in the BGO detector, which means that both detectors have signal, the event is rejected. Otherwise, if only the HPGe detector has signal, is stored. This technique is called "Compton suppression spectrometry". The BGO detector is also called "anti-Compton" detector. A schematic view of the Compton suppression is given in Fig. 3

The JUROGAM II array

The JUROGAM II array [START_REF] Nolan | Large arrays of escape-suppressed gamma-ray detectors[END_REF], a large -ray spectrometer consisting of 24 clover and 15 coaxial tapered detectors placed at the target position, has been used to detect prompt -rays. The clover detectors were placed on two rings at 75.5 (12 clovers) and 104.5 (12 clovers) symmetric with respect to 90 . The tapered detectors were also placed on two rings at 133.6 (10 detectors) and 157.6 (5 detectors). All Ge detectors of the array are equipped with Compton-suppression shields made of BGO and Hevimet collimators in front of the detectors to improve the P/T ratio. The add-back method of summing the coincident -ray energies in separate crystals in the same clover detector within a 200 ns time window was used in this work. For a -ray with the energy of 1.33 MeV, the full JUROGAM spectrometer with 39 detectors has a total photo-peak efficiency around 5% and energy resolution of 3.5 keV (FWHM) [START_REF] Nolan | Large arrays of escape-suppressed gamma-ray detectors[END_REF]. The schematic configuration of the JUROGAM II detector array is shown in Fig. 3.5.

The RITU gas-filled recoil separator

Recoil separators for heavy-ion induced fusion products have become important tools to separate the fusion-evaporation residues from the primary beam and other unwanted reaction products, and then transport the recoils to the focal plane for further measurements (as for example, decay properties of heavy elements, search for long-lived isomeric states in nuclei). The Recoil Ion Transport Unit (RITU) is a gas-filled recoil separator [START_REF] Leino | Gas-filled recoil separator for studies of heavy elements[END_REF][START_REF] Sarén | Absolute transmission and separation properties of the gas-filled recoil separator ritu[END_REF][START_REF] Leino | In-flight separation with gas-filled systems[END_REF], which is coupled to the JUROGAM II array, as is schematically shown in Fig. 3.6.

The RITU gas-filled recoil separator

The typical separator is of the type DQQ with a magnetic dipole followed by a quadrupole doublet to focus the fusion products into the focal plane, the reason being that separation of primary beam immediately downstream from the target reduce the background at the focal plane. RITU is based on a standard DQQ magnetic configuration, it consists of a total of four focusing components that are in a Q 1 DQ 2 Q 3 configuration. The first magnetic quadrupole (Q 1 ) is vertically focusing to improves the dipole magnet acceptance, which leads to an increase of 30% in the angular acceptance [START_REF] Leino | Gas-filled recoil separator for studies of heavy elements[END_REF]. The dipole magnet (D) is used to separate ions depending on their different magnetic rigidities B⇢. The Q 2 and Q 3 quadrupole magnets are horizontally and vertically focusing, respectively.

The relationship between the magnetic rigidity B⇢ and the average charge state q ave of ions of a nuclear species Z is given by the formula [START_REF] Ghiorso | Sassy, a gas-filled magnetic separator for the study of fusion reaction products[END_REF]

B⇢ = p q ave = mv (v/v 0 )Z 1/3 = 0.0227A/Z 1/3 T m, (3.8)
where B is the magnetic field strength, ⇢ is the curvature radius, p and mv is the momentum of the reaction product and v o is the Bohr velocity = c/137 m/s = 2.19 ⇥ 10 6 m/s, q ave = (v/v o )Z 1/3 is approximate and is given by the Thomas-Femi model of the atom. Eq. 3.8 can be used for a rough estimate of the degree of separation residues. In heavy element studies, helium and hydrogen are the practical choices for filling gas due to their favourable scattering properties. The choice between these alternatives may be dictated by background suppression properties, safety considerations, or by the fact that higher field strengths are required in case of hydrogen and very asymmetric reactions [START_REF] Ts Oganessian | Zeitschrift für Physik D Atoms[END_REF]. The internal volume of the RITU is filled with a flow of helium gas with the pressure around 1 mbar.

Usually, pressures around 1 mbar lead to the minimum focal plane image size and also to increase of the recoil transmission.

The GREAT spectrometer

Gamma Recoil Alpha Tagging (GREAT) [START_REF] Page | The GREAT spectrometer[END_REF] spectrometer is placed at the focal plane of RITU separator. It is designed to measure the decay properties of reaction products transported to the focal plane of the recoil separator. In addition, GREAT can either be employed as a sensitive stand-alone device for decay measurements at the focal plane, or used to provide a effective tag for prompt conversion electrons or rays measured with arrays of detectors deployed at the target position [START_REF] Page | The GREAT spectrometer[END_REF].

The GREAT detector array is composed of several detector types: multiwire proportional counter (MWPC), double-sided silicon strip detectors (DSSDs), PIN diodes, planar germanium detector and three clover detectors. A schematic view of the GREAT spectrometer is shown in Fig. 3.7. The MWPC is placed at the entrance of GREAT, which can measure the position of the recoils and deliver the time reference for the delayedcoincidences, and for the time of flight (TOF) of the recoils between the MWPC and the DSSDs.

The recoils are implanted into DSSDs, which are used to measure the energies of ions, of subsequent emitted ↵ particles, particles and protons. Each DSSD has an active area of 60 mm ⇥ 40 mm and a thickness of 300 µm. The strip pitch is 1 mm in both directions, matching the position resolution of the MWPC and giving a total of 4800 pixels that can be used for correlation analysis. Reaction products are typically implanted into the DSSDs at depths of around 1-10 µm, depending on the target-projectile combination [START_REF] Page | The GREAT spectrometer[END_REF].

An array of silicon PIN photodiode detectors to measure conversion electrons and ↵ particles that escape from the DSSDs in the backward direction.

Total-Data Readout (TDR)

The active area of each detector element is 28 mm ⇥ 28 mm with a thickness of 500 µm.

Behind the DSSD, a planar Ge strip detector is mounted, which is used to for the measurement of X-rays and low-energy rays. The active area of each detector element is 120 mm ⇥ 60 mm with a thickness of 15 µm. The strip pitch on both faces is 5 mm, providing position information that can be correlated with other GREAT detectors. It can also be used to detect highenergy particles ( 2 MeV) that penetrate through the DSSDs [START_REF] Page | The GREAT spectrometer[END_REF].

For the measurement of high-energy -rays, a set of three clover detectors have been placed around the focal plane reaction chamber. Each of the four crystals has a diameter of 70 mm and a length is of 105 mm. The detectors are surrounde by BGO suppression shields to improve their P/T ratio.

Total-Data Readout (TDR)

In general, nuclear experiments are using data-acquisition systems which are triggered by an event in a pre-defined detector. The main problem of a conventional trigger method is that the dead-time causing a loss of events and the data written to tape include both good and random data, which later must be separated. The random data also increase the readout time and the data rate to tape.

To overcome dead-time limitations due to reading all the data, the one of the solution is time stamping them, and then collecting the event fragments together in software in the event builder using spatial and temporal correlations. Total-Data Readout (TDR) [START_REF] Lazarus | The GREAT triggerless total data readout method[END_REF] data-acquisition systems are needed. In TDR acquisition systems all channels run independently and are associated in software to reconstruct events. The correlation in the event builder can be simple. For example, in order to eliminate random rays measured with JU-ROGAM II at the target position, one can tag with recoils, and then looking back in time using a window centred on the time of flight [START_REF] Lazarus | The GREAT triggerless total data readout method[END_REF].

The most important and valuable feature of the TDR is that it is triggerless and the data is timestamped from a global 100-MHz clock giving a time resolution of 10 ns. The data from TDR are not structured or filtered, apart from the time-ordered, it allows the offline process data more flexible and events reconstructed in real time in the event builder using temporal and spatial associations depend on the physics of the experiment.

In the experiment of the present work, all the data were recorded by the triggerless TDR data acquisition and the events were time-stamped using a 100-MHz clock. The data acquisition system employs an OR gate in the GREAT spectrometer as a software trigger. It requires a signal in any of the GREAT detectors to be recorded for an event to start, and JUROGAM II data are only stored if they arrive within a give time window relative to the signal in GREAT. The data from the JUROGAM II detectors were buffered for 5 µs if there was a signal at the focal plane, and provided a JUROGAM II detector fired within the time period preceding the signal in GREAT, the data was stored to disc [9]. A typical time structure for each detector relative to the event stamp in the DSSDs (T=0) is presented in Fig. 3.8. In this case, data would be stored on disc if a signal within the trigger width of 3 µs, and if it is registered 1 µs before or 2 µs after an event in the DSSDs. There is no limit to choose the trigger or delay widths due to the signals are collected with a 10 ns time stamp, except for the the upper limit of 5 µs resulting from the software trigger delay. Although there is a maximum correlation time of 5 µs for each event, there is no limitation in the correlation time between separate events [9]. The data processed using the GRAIN [START_REF] Rahkila | Grain-a java data analysis system for total data readout[END_REF] software package will be discussed in the next chapter.

Chapter 4

Experimental details and processing of the data

Experimental details

The experiment was performed at the University of Jyväskylä, Finland, using the JUROGAM II + RITU + GREAT setup. High-spin states in 135,136 Nd were populated using the fusion-evaporation reaction 100 Mo( 40 Ar, xn). The 40 Ar ion beam was produced in an ECR ion source and accelerated by the K130 cyclotron to final energy of 152 MeV, the beam intensity was around 14 pnA. One week beam time was allocated for this experiment. We used as target a self-supporting enriched 100 Mo foil of 0.5 mg/cm 2 thickness. The 135 Nd and 136 Nd nuclei were the most strongly populated in the reaction, with cross sections of around 100 mb each, for a total reaction cross section calculated with PACE4 of 480 mb.

Data processing

In this experiment ⇡ 4000 Gb of data spread across ⇡ 2000 files were stored. A total of 5.1 ⇥ 10 10 prompt -ray coincidence events with fold 3 were collected. All the data were recorded by the triggerless TDR data acquisition system and the events were time-stamped using a 100 MHz clock with the time resolution of 10 ns.

Energy calibrations and gain matching

During the experiment, the gain of each detector could drift. Hence, it is important to calibrate (establish the relation ADC channel-energy) each detector and to gain match all detectors. In this work, the JUROGAM II detectors, the focal plane planar, and the clover detectors were calibrated before and after the experiment using the standard -ray sources 152 Eu and 65 66 4.2. Data processing 133 Ba. To calibrate the detectors a polynomial expression was used of the form

E = a + bx + cx 2 , (4.1)
where E is the -ray energy, x is the ADC channel number associated with the centroid position of the photopeak; a, b, and c are coefficients of the polynomial used for obtaining the calibrated energy.

In the present work, we used a program ENCAL from the radware [START_REF] Radford | {ESCL8R} and levit8r: Software for interactive graphical analysis of {HPGe} coincidence data sets[END_REF][START_REF] Radford | Background subtraction from in-beam {HPGe} coincidence data sets[END_REF] analysis package to obtain the coefficients a, b, and c. Fig. 4.1 shows a sample of overlayed some Ge detectors of JUROGAM II array after calibration.

In addition, each strips of the DSSDs was calibrated using a combination of 239 Pu, 241 Am, and 244 Cm sources.

Efficiency calibrations for Ge detectors

In order to make the correction of the measured -ray intensities for the detection efficiency of the JUROGAM II array, we used the program EFFIT from the radware [START_REF] Radford | {ESCL8R} and levit8r: Software for interactive graphical analysis of {HPGe} coincidence data sets[END_REF][START_REF] Radford | Background subtraction from in-beam {HPGe} coincidence data sets[END_REF] analysis package.The efficiency of the detectors was obtained by fitting the data points using the equation,

ln(✏) = [(A + Bx + Cx 2 ) G + (D + Ey + F y 2 ) G ] 1/G , (4.2) with x = ln( E 100 ), y = ln( E 1000 ), (4.3)
Chapter 4. Experimental details and processing of the data 67 where ✏ is the efficiency and E is the -ray energy in keV. The A, B, and C parameters are used to control the low energy region efficiency curve. The parameter C is often fixed to 0, and D, E, and F are the parameters that are used to adjust the high energy part of the efficiency curve. G is the parameter which determines the shape of the region between the high and low energy efficiency curves. In present experiment, the following parameters were used to obtain the best fit of JUROGAM II efficiency: A = 6.2, B = 1.8, C = 0.0, D = 5.5, E = 0.6, F = 0.0 and G = 20.0.

Doppler shift correction

For fusion-evaporation reactions, the -rays emitted by a in-flight recoil nuclei are Doppler-shifted when a stationary detector is used. The line width is also broadened due to the Doppler effect, hence, it is necessary to Doppler correct the -ray energies. The shifted energy is determined by the formula:

E 0 = E p 1 2 1 cos✓ , (4.4)
where E is the unshifted -ray energy, = v c , v is the velocity of the recoil, c is the speed of light, and the detector angle ✓ is defined with respect to the beam direction.

If << 1, the shifted energy is calculated using the following formula:

E 0 = E (1 + cos✓). (4.5)
In present work, the best resolution was obtained for a value = 0.0245 which was therefore used to correct the Doppler shift of all rays detected in JUROGAM II.

Add-back for the clover detectors

The clover HPGe detector consists of four crystals. As the four crystals are close-packed and put in a single cryostat, there is the possibility that -rays Compton scatter from one crystal to another. In a Compton scattering event it can happen that the -ray deposits part of its energy in one crystal and then enter a neighbouring crystal to deposit a part or the remainder of its energy. In practice, the energy signals of the crystals that hit simultaneously the crystals are recorded in an event-by-event mode which are time correlated [START_REF] Duchêne | The Clover: a new generation of composite ge detectors[END_REF]. Thus, these events can be used to reconstruct the full -ray energy by summing the individual events in a procedure called "add-back" technique. The main advantage of the clover HPGe detector with add-back is the increase of the photo-peak efficiency (P/T) ratio.

Spin and parity assignments

After preparing the data as described above, we sorted all raw data using the GRAIN analysis package [START_REF] Rahkila | Grain-a java data analysis system for total data readout[END_REF]. Fully symmetrized, three-dimensional (E -E -E ) coincidence cubes were analyzed using the levit8r program from the radware analysis package [START_REF] Radford | {ESCL8R} and levit8r: Software for interactive graphical analysis of {HPGe} coincidence data sets[END_REF][START_REF] Radford | Background subtraction from in-beam {HPGe} coincidence data sets[END_REF]. The program allows for a double gate to be set on two mutual coincident transition energies. The double gate will produce a slice on the third axis of the cube showing a one-dimensional -ray spectrum with all the detected -rays which are in coincidence with the two -ray transitions that were selected in the double gate. Then, we constructed the level scheme of the two nuclei 135 Nd and 136 Nd. The multipolarities of the -rays were extracted using the Directional Correlation from Oriented states (DCO) ratios (R DCO ) and two-point Angular Correlation (anisotropy) ratios R ac [START_REF] Krämer-Flecken | Use of dco ratios for spin determination in -coincidence measurements[END_REF][START_REF] Chiara | Probing sd -fp cross-shell interactions via terminating configurations in 42,43[END_REF], as is described in the following.

(i) DCO ratios In the heavy-ion evaporation reaction, the angular momenta of the compound nucleus are aligned in a plane perpendicular to the beam direction. The angular correction of -rays emitted from oriented states depends on the spins of the involved levels. Thus, we can deduce the DCO values and assign the spin of one of the two levels linked by a given -ray if the spin of the other level is known. In the DCO method, the ratio between the intensities of two coincident -rays is analyzed. The angles between the two -rays and the beam direction are ✓ 1 and ✓ 2 , respectively. A third angle is also used to describe the angular position of the detectors. The intensity of the transition

R DCO = W (✓ 2 , ✓ 1 , ) W (✓ 1 , ✓ 2 , ) . (4.6)
The corresponding experimental DCO ratio is then written as

R DCO = I 2 ✓ 1 (gated by 1 at ✓ 2 ) I 2 ✓ 2 (gated by 1 at ✓ 1 )
.

(4.7)

The DCO ratios depend on the angles ✓ 1 , ✓ 2 , and . In this experiment, JU-ROGAM II detectors are placed in four rings, as shown in Fig. 3.5. To extracted the DCO ratios from the ring of detectors placed at different ✓ angles, we integrated over all angles . Then, the R DCO values were extracted from -matrices, which were formed by sorting prompt coincidence events with the detectors at (157.6 ) versus those at (75.5 and 104.5 , corresponding to Note that the DCO ratios depend upon the type of transition gated on, and the measured transition DCO ratio will invert if exchange of the angles or of the gating.

(ii) Angular correlations ratio (R ac )

In addition of the DCO analysis, -ray multipolarities can also be obtained by measuring the two-point angular correlations ratios (R ac ), which is calculated by extracting the intensity of 1 in the detectors at ✓ 1 from a gate on 2 in all detectors and the intensity of 1 in the detectors at ✓ 2 from a gate on 2 in all detectors. The angular correlation ratio is

R ac = I 1 ✓ 1 (gated by 2 at all angles) I 1
✓ 2 (gated by 2 at all angles) . (4.9)

The R ac method has some advantages, i.e., the value is independent of the gating transition character, can distinguish between I = 2 pure quadrupole and I = 1 pure dipole transitions. The two-point angular correlation ratio 4.2. Data processing is very useful for weak transition, which is the case of many new transitions identified in the 135 Nd and 136 Nd.

In the present data analysis, the values of R ac were extracted frommatrices, which were formed by sorting prompt coincidence events with (133.6 and 157.6 ) versus (all angles) and (75.5 and 104.5 ) versus (all angles) combinations, by setting the same energy gates on the (all angles) projection spectrum in both matrices, and projecting on the another axis. Then, the R ac ratio was calculated using the extracted intensities of the -rays of interest (I ) from these spectra, normalized by the different efficiency of the two sets of detectors. The R ac is calculated using the formula The typical values for stretched dipole and quadrupole transitions are around 0.8 and 1.4, respectively.

It should be noted that both R DCO and R ac methods can only be used to deduce I, but not their parity. To determine the electric or magnetic character of the emitted -rays, linear polarization measurements are required. A detailed description of the method applied to arrays of HPGe detectors can be found in Ref. [START_REF] Starosta | Experimental test of the polarization direction correlation method (PDCO)[END_REF].

Chapter 5

Towards complete spectroscopy of 136

Nd

In this chapter the experimental results obtained from the in beamspectroscopy study of the even-even nucleus 136 Nd are presented. The structure of the various bands is discussed in the frameworks of the constrained and tilted axis cranking covariant density functional theory, the multi-j shell particlerotor model, and the cranked Nilsson-Strutinsky model.

Experimental results and level scheme

The nucleus 136 Nd located near the center of the triaxial A = 130 region which has been previously investigated both in experimentally [START_REF] Habs | Quasirotational groundstate bands of eveneven nuclei with 50[END_REF][START_REF] Aléonard | Experimental study of 136 Nd at high angular momentum[END_REF][START_REF] Barci | Effective moment of inertia in 132 Ce, 134 Nd and 136 Nd[END_REF][START_REF] Paul | Band crossing in the gamma-soft nucleus 136 Nd[END_REF][START_REF] Billowes | Lifetime and g-factor measurements of yrast states in 134 Nd and 136 Nd[END_REF][START_REF] Beck | Superdeformed bands in Nd nuclei[END_REF][START_REF] Clark | A pair of identical superdeformed bands in 136 Nd[END_REF][START_REF] Petrache | Competing mechanism for generating high spin excitations in gamma-soft nuclei: the 136 Nd case[END_REF][START_REF] Petrache | Multiple dipole bands at prolate shapes in 136 Nd and their description through the projected shell model[END_REF][START_REF] Mergel | Candidates for chiral doublet bands in 136 Nd[END_REF][START_REF] Saito | Yrast and non-yrast 2+ states of 134 Ce and 136 Nd populated in relativistic coulomb excitation[END_REF][START_REF] Perries | Decay-out of the yrast superdeformed band in 136 Nd: Towards an experimental extraction of the neutron pairing gap at large deformation[END_REF] and theoretically (see Refs. [START_REF] Long | Collective backbending effect in the interacting boson model[END_REF][START_REF] Vretenar | Interacting boson model plus broken-pair description of high-spin dipole bands[END_REF][START_REF] Tanabe | Microscopic properties of the superdeformed rotational states in light rare-earth nuclei 132 Ce and 134,136 Nd[END_REF][START_REF] Próchniak | Collective quadrupole excitations in the 50 < Z, N < 82 nuclei with the general bohr hamiltonian[END_REF][START_REF] Sheikh | Multiquasiparticle -band structure in neutron-deficient Ce and Nd isotopes[END_REF]). However, most of the experimental results were reported more than twenty years ago, being obtained with less efficient -ray arrays, which allowed to establish a limited level scheme.

In the present work, the level scheme of 136 Nd has been extended significantly. Many new bands have been identified both at low and high spin, among which five nearly degenerate bands which are interpreted as chiral partners. In addition of the excitation energies, the spins and parities of several previously known bands are revised and firmly established.

The level scheme of 136 Nd splitted into four partial level schemes is presented in Figs. 5.1, 5.6, 5.14 and 5.15. Most of the observed transitions were grouped in bands: the band, two bands of negative parity at low spins (N1-N2), nine bands at medium spins (L1-L9), six bands of dipole transitions (D1-D6) and five chiral partner bands (D1-chiral, D2-chiral, D3-chiral. D4-chiral, D5-chiral), four high-spin bands (T1-T4) and five highly-deformed bands (HD1-HD5). In addition, many new states have been newly identified and some previous states reported have been revised. The experimental information on the observed transitions is given in Table 5.1.

The low-spin band and bands N1, N2

The band was first reported up to I ⇡ = 5 + [START_REF]ENSDF database[END_REF], which now is extended to higher spins and linked by the 815-and 600-keV transitions to another previously known band built on the 10 + state at 3768 keV [START_REF] Paul | Band crossing in the gamma-soft nucleus 136 Nd[END_REF]. The band is now observed up to I ⇡ = 20 + : the previously reported 895-keV transition on top of the band has been replaced by the 1018-keV transition. The new transitions of 490, 735, 907, 1163, 1410.7, 1410.8, and 1429 keV have been identified connecting the band to the positive-parity bands L1 and L2, and to the 16 + state deexcited by the 1093-keV transition. Coincidence spectra obtained by double-gating on selected transitions of the band are shown in Fig. 5.2.

For band N1, we newly identified the 1077-, 410-, 1079-, 744-and 832-keV transitions which were placed on the top of the previously known states of band N1. In addition, we observed two new transitions of 381 and 612 keV from the 10 + state of the GSB towards the intermediate 9 state, which in turn decays to the 7 and 8 states of band N1, and directly towards the 9 state of band N1, respectively.

The new 6 + state at 2228 keV has been identified, which decays to the ground-state band (GSB), to the band, and to band N1 via the 1252-keV, 686-keV and 192-keV transitions, respectively. The newly observed band N2 decays to the new 6 + state, to the band, and to band N1 via the 366-keV, 300-keV and 154-keV transitions, respectively, and its 6 and 7 states are fed by the 412-keV and 164-keV transitions from band N1. The bandhead spin of the band N2 is fixed by the I = 1 transitions with energies of 366 and 300 keV. The negative parity of band N2 is fixed by the 412-keV E2 transition connecting the 8 state of band N1 to the 6 state of band N2. The negativeparity assignment to band N2 is further supported by the low-energy 154and 164-keV transitions of M 1/E2 character between bands N1 and N2. The double-gated spectra are given in Fig. 5.3 show the transitions of band N2. Band L3, previously reported in Ref. [START_REF] Petrache | Competing mechanism for generating high spin excitations in gamma-soft nuclei: the 136 Nd case[END_REF], is confirmed. One new transition of 740 keV connecting the 19 + state with the 18 + state of band L1 has been identified. been added to band L8. Band L9 is newly identified. It decays to band L7 via the 912-keV transition and its connected to band D2 via the 318-and 744-keV transitions.

Bands T1, T3 and T4 previously reported in Ref. [START_REF] Petrache | Competing mechanism for generating high spin excitations in gamma-soft nuclei: the 136 Nd case[END_REF] are confirmed, and several new transitions have been identified. A new relatively strong E2 transition of 1121 keV has been added on top of band T1 (see Fig. 5.1). The energies of the highest in-band transitions of bands T3 and T4 have been revised to 1391 and 1323 keV, respectively, whereas the spin and parity of the highest two and three levels, respectively, are tentative.

Band T2 is newly identified. It is built on the 15 state and decays only to band T1 via the 1079-keV transition. It develops up to very high spin [START_REF] Tonev | Candidates for twin chiral bands in 102 Rh[END_REF]. A sum of doubly-gated spectra on the in-band transitions of band T2 is given in Fig. 5.5.

The dipole bands

A particular feature of the level scheme of 136 Nd is the existence of five pairs of nearly degenerate bands (D1 and D1-chiral, D2 and D2-chiral, D3 and D3-chiral, D4 and D4-chiral, D5 and D5-chiral) and the dipole band D6 without partner. The partial level scheme of 136 Nd showing the five pairs of chiral bands and band D6 is given in Fig. 5.6. One can observe that all these dipole bands decay to the yrast partners via high-energy quadrupole transi- tions which, due to the E 5 dependence, are stronger that the in-band dipole transitions which have low energy and E 3 dependence, resulting in a factor of 10 to 20 in favor of the connecting transitions. This is the one of the reasons why many in-band dipole transitions of the non-yrast bands could not be observed with the present statistics. Band D1, which was first reported in Ref. [START_REF] Petrache | Multiple dipole bands at prolate shapes in 136 Nd and their description through the projected shell model[END_REF], is confirmed by the present study, but the spins and parity are revised. Three new crossover transitions of 639, 756 and 865 keV, and three new transitions of 485, 594 and 613 keV placed on top of the previously reported band, have been identified. Nine new decay-out transitions with energies of 329, 479, 583, 845, 905, 980, 1065, 1203 and 1234 keV have been identified, leading to a band-head of I ⇡ = 11 + , with spin 2h higher than that previously spin. The assigned positive parity is based on a series of the relatively strong connecting transitions to low-lying states. The new band D1-chiral decays to band D1 via high-energy E2 transitions ( 716 keV, 840 keV, 515 keV, 1019 keV, and 1045 keV). Double-gated spectra showing the transitions connecting one of the newly identified band D1-chiral to band D1 are given in Fig. 5.7.

Band D2 first reported in Ref. [START_REF] Petrache | Multiple dipole bands at prolate shapes in 136 Nd and their description through the projected shell model[END_REF] is confirmed, with one new crossover Band D5 first reported in Ref. [START_REF] Petrache | Multiple dipole bands at prolate shapes in 136 Nd and their description through the projected shell model[END_REF] is confirmed. Several new decay-out transitions with energies of 180, 412, 709, 765, 952, 1480 and 1660 keV have been identified. Band D5-chiral consisting of in-band dipole and E2 crossover transitions is new. A sum of coincidence spectra obtained by double-gating on the transitions of band D5 is given in Fig. 5.11. Band D5-chiral is linked to band D5 by many dipole and quadrupole transitions. The relatively large intensities of the transitions of bands D5 and D5-chiral, made possible the extraction of the B(M1)/ B(E2) branching ratios for both bands, which are very similar and strongly support the chiral interpretation [START_REF] Petrache | Evidence of chiral bands in even-even nuclei[END_REF].

Band D6 first reported in Ref. [START_REF] Petrache | Multiple dipole bands at prolate shapes in 136 Nd and their description through the projected shell model[END_REF] is confirmed. In the present work, we extended it up to I ⇡ = 30 by adding on top of the previous band three dipole transitions of 581, 604 and 660 keV, and four crossover transitions of 945, 1112, 1185 and 1264 keV.

The highly-deformed bands

Five highly-deformed (HD) bands have been observed in 136 Nd in the present work. The three bands HD2, HD4 and HD5 are new. All bands are in coincidence with known low-lying transitions of 136 Nd. However, no linking transitions to the normal-deformed states have been found. Tentative spins, parities and excitation energies have been assigned based on the CNS model calculations (see the following sections). Spectra constructed by doubly-gating on all in-band transitions for the five bands are given in Figs. 5.12 and Fig. 5.1 -Experimental information including the -ray energies, energies of the initial levels E i , intensities I , anisotropies R DCO and or R ac , multipolarities, and the spin-parity assignments to the observed states in 136 Nd. The transitions listed with increasing energy are grouped in bands. The deduced values for R DCO with a stretched quadrupole gate are ⇡ 1 for stretched quadrupole and ⇡ 0.46 for dipole transitions, while the ratio is close to 1 for a dipole and 2.1 for a quadrupole transition when the gate is set on a dipole transition. The R ac values for stretched dipole and quadrupole transitions are ⇡ 0.8 and ⇡ 1.4. 136 Nd.

E a E i (keV) I b R DCO c R ac d Mult. J ⇡ i ! J ⇡ f GSB 373
E E i (keV) J ⇡ i ! J ⇡ f 5.2. Discussion

Discussion

As mentioned above, a particularly important feature of level scheme of 136 Nd is the observation of the five pairs of doublet bands (see Fig. 5.6). These five pairs of doublet bands is the most extended set of chiral bands observed until now, and therefore represent an important support for the existence of the multiple chiral doublet (M D) phenomenon. The constrained and tilted axis cranking covariant density functional theory (TAC-CDFT) and particle-rotor model (PRM) calculations were performed to describe the identified doublet bands [55,[START_REF] Petrache | Evidence of chiral bands in even-even nuclei[END_REF]. In addition, all rotational bands of 136 Nd were discussed within the cranked Nilsson-Strutinsky (CNS) framework [START_REF] Lv | Evolution from -soft to stable triaxiality in 136 Nd as a prerequisite of chirality[END_REF]. In this part of the thesis, the results of these calculations are presented.

M D interpretation of the chiral bands within TAC-CDFT framework

Generally speaking, in oder to describe the chiral rotations, three-dimensional tilted axis cranking covariant density functional theory (3D TAC-CDFT) is required, which has been used to investigate the chiral modes of 106 Rh [START_REF] Zhao | Multiple chirality in nuclear rotation: A microscopic view[END_REF]. For the doublet bands built on the 4-and 6-quasiparticle configurations identified in 136 Nd, the 3D TAC-CDFT calculations are very challenging. For simplicity, the observed five pairs of rotational bands are investigated within the two-dimensional framework of TAC-CDFT [START_REF] Peng | Covariant density functional theory for magnetic rotation[END_REF][START_REF] Zhao | Novel structure for magnetic rotation bands in 6 0Ni[END_REF][START_REF] Meng | Progress on tilted axis cranking covariant density functional theory for nuclear magnetic and antimagnetic rotation[END_REF][START_REF] Meng | Progress on tilted axis cranking covariant density functional theory for nuclear magnetic and antimagnetic rotation[END_REF]. However, the chiral nature of the assigned configurations was tested by performing 3D TAC-CDFT calculations for one of chiral doublet bands D3 as an example. In calculations, the relativistic density functional PC-PK1 [START_REF] Zhao | New parametrization for the nuclear covariant energy density functional with a point-coupling interaction[END_REF] is adopted and the Dirac equation is solved in a 3D harmonic oscillator basis in Cartesian coordinates with 10 major shells which provide convergent results in TAC-CDFT calculations [START_REF] Meng | Progress on tilted axis cranking covariant density functional theory for nuclear magnetic and antimagnetic rotation[END_REF]. The contribution of the pairing correlations has also been investigated, concluding that they negligible, in the discussed multi-quasiparticle configurations: the changes of total energy and total angular momentum at the rotational frequency 0.2 MeV are within 0. 005% and 4.5% for configuration A. Thus, the pairing correlations be safely neglected within the TAC-CDFT framework [START_REF] Petrache | Evidence of chiral bands in even-even nuclei[END_REF].

In oder to understand the nature of the observed band structure in 136 Nd, by minimizing the energy with respect to the deformation , both adiabatic and configuration-fixed -constrained calculations similar to those in Ref. [START_REF] Meng | Possible existence of multiple chiral doublets in 106 Rh[END_REF] were performed to search for the possible configurations and deformations. The results are provided in Table 5.3. One can see that three positive-parity configurations (labels A, B, and C) and five negative-partiy configurations (labels D, D ⇤ , E, F, and H) are candidates for the observed nearly degenerate partner bands. One can also see that all these configurations with particle-hole excitations possess remarkable triaxial deformation, which is the typical feature for the chiral rotational bands. Subsequently, the assignment of configura-tions listed in Table 5.3 were justified based on the quasiparticle alignments. Furthermore, the configurations and deformations of the assigned configurations were reexamined by TAC-CDFT calculations. The obtained results in Table 5.3 were also used to calculate the B(M 1)/B(E2) ratios.

) 2 ⌦ ⌫(h 11/2 ) 1 (s 1/2 d 3/2 ) 1 (⌦ ⇠ + 3 2 ) D ⇤ 1.346 (0.21, 22 ) ⇡(h 11/2 ) 2 ⌦ ⌫(h 11/2 ) 1 (s 1/2 d 3/2 ) 1 (⌦ ⇠ 3 2 ) E 1.937 (0.21, 23 ) ⇡(h 11/2 ) 2 (d 5/2 g 7/2 ) 2 ⌦ ⌫(h 11/2 ) 1 (s 1/2 d 3/2 ) 1 F 2.778 (0.20, 35 ) ⇡(h 11/2 ) 1 (d 5/2 g 7/2 ) 1 ⌦ ⌫(h 11/2 ) 2 H 3.494 (0.20, 37 ) ⇡(h 11/2 ) 1 (d 5/2 g 7/2 ) 3 ⌦ ⌫(h 11/2 ) 2
Fig. 5. 16 shows the calculated excitation energies and h! versus spin I for the bands D1-D5 using the TAC-CDFT formalism, in comparison with the experimental data. One can see that the theoretical results based on the assigned configurations agree with experimental data, the chiral partner having similar behavior. In detail, for band D1, the theoretical results based on configuration A agree with experimental data well, the configuration assignment being thus validated. Meanwhile, no other configuration can be found to reproduce band D1-C, which indicates that bands D1 and D1-C are based on the same intrinsic state, , and therefore that the bands D1 and D1-C are good candidates for chiral partner bands; Similarly, one can conclude that D2 and D2-C, D5 and D5-C, D3 and D3-C, as well as D4 and D4-C are good candidates of chiral doublets bands built on the same intrinsic configurations B, C, D ⇤ , and D, respectively. This is one of the evidences in favor of the chiral interpretation.

It should be pointed that the energy references for the positive-and negative-parity calculated bands are taken as those of band D1 at I = 11 h and of band D4 at I = 15 h, respectively. One should note that the backbending phenomenon observed in bands D2, D3, and D4 can be due to crossing with 8-, 6-, and 6-qp configurations, respectively, which is beyond the current TAC-CDFT calculation.

The quasiparticle alignments calculated by TAC-CDFT are shown in Fig. 5.17 in comparison with the experimental data. The parameters J 0 = 11 h2 MeV 1 and J 1 = 20 h4 MeV 3 for the Harris formula J = J 0 + J 1 ! 2 have been adopted. It is clearly seen that the calculated results based on the assigned configurations reproduce the experimental data very well. In oder to give more strongly theoretical support for the chiral interpretation of the observed doublet bands in 136 Nd, our collaborators performed challenging calculations based on 3D TAC-CDFT. Here, we only use band D3 as an example to show the calculated results. In 3D TAC-CDFT calculations, the D ⇤ configuration was assigned to the band D3.

Discussion

M D interpretation of the chiral bands within PRM framework

To investigate in detail to what extent the chiral geometry of the observed doublet bands in 136 Nd is realized, a particle-rotor model (PRM) which couples nucleons in four single-j shells to a triaxial rotor core has been developed [55].

In the performed PRM calculations, the unpaired nucleon configurations are consistent with those in TAC-CDFT used for the doublet bands D1-D5, and the corresponding quadrupole deformation parameters ( , ) are obtained from triaxial constrained CDFT calculations [START_REF] Meng | Possible existence of multiple chiral doublets in 106 Rh[END_REF]. The moments of inertia J 0 and Coriolis attenuation factors ⇠ are adjusted to reproduce the trend of the energy spectra. The corresponding details are listed in Table 5.4. In addition, for the electromagnetic transitions, the empirical intrinsic quadrupole moment

Q 0 = (3/ p 5⇡)R 2 0 Z
, and gyromagnetic ratios for rotor g R = Z/A and for nucleons g p(n) = g l + (g s g l )/(2l + 1) (g l = 1(0) for protons (neutrons) and g s = 0.6g s (free)) [3] are adopted. ,D5-c) the energy differences between the doublet bands decreasing with increasing spin and finally merge together. These rather small energy differences between doublet doublets support their chiral interpretation.

The staggering parameters S(I), is one of the more sensitive parameters to examine the energy spectra. The standard fingerprints for chiral bands outlined in Ref. [START_REF] Vaman | Chiral degeneracy in triaxial 104rh[END_REF] require that S(I) is independent of spin. One can see that the PRM results are in impressive good agreement with the experimental values. In detail, for band D1 and D6, one observes that the decreasing B(M 1)/B(E2) values with increasing spin are well reproduced by model. For band D2, one observes that the B(M 1)/B(E2) values firstly increase and then an abruptly decrease at I = 21h. To understand it, after analyzing the corresponding PRM wave function, we found that, at I < 20h, the largest component of the state is I s ⇠ I (I s the angular momentum component along the short axis), while for I 20h, the largest one is I s ⇠ I 2. This structure change causes the small B(M 1) value at I = 20h and small B(E2) values at I = 20 and 21h, and hence large B(M 1)/B(E2) value at I = 21h. For bands D3 and D4, one observes a similar behavior of the B(M 1)/B(E2) values. It seems that they are M D built on identical configuration as in 103 Rh [START_REF] Kuti | Multiple chiral doublet bands of identical configuration in 103 Rh[END_REF]. However, as their spectra are interweaved Table 5.4 -The parities, unpaired nucleon configurations, quadrupole deformation parameters ( , ), moments of inertia J 0 (unit h2 /MeV), and Coriolis attenuation factors ⇠ used in the PRM calculations for bands D1-D6 and their partners.

Band Parity

Unpaired nucleons ( , ) each other at several spins, this possibility is excluded. In the calculations, ⇡(1h 11/2 ) 2 ⌦ ⌫(h 11/2 ) 1 (2d 3/2 ) 1 configuration with three single-j shells is used to describe D3 and ⇡(1h 11/2 ) 2 (2d 5/2 ) 2 ⌦⌫(h 11/2 ) 1 (2d 3/2 ) 1 configuration with four single-j shells is used to describe D4, respectively. As shown in Fig. 5.22, we can observe that the present PRM calculations do not agree very well with the data of D3, while for band D4, the calculated results reproduce very well the experimental data for I 19h. For D5 and D5-C, their B(M 1)/B(E2) values are quite similar and in very good agreement with the PRM calculations fulfilling thus the characteristics of chiral doublet bands [51,[START_REF] Vaman | Chiral degeneracy in triaxial 104rh[END_REF].

J 0 ⇠ D1 + ⇡(1h 11/2 ) 1 (2d 5/2 ) 1 ⌦ ⌫(1h 11/2 ) 1 (2d 3/2 )
The great successes in reproducing the energy spectra and B(M 1)/B(E2) ratios for the doublet bands in 136 Nd motivate us to examine the angular momentum geometries in PRM for the observed doublet bands. To exhibit their chiral geometry, we calculated the expectation values of the squared angular momentum components along the intermediate (i-), short (s-), and long (l-) axes for the rotor, valence protons, and valence neutrons. The obtained results of bands D2, D4, and D5 are shown in Figs. 5.23, 5.24, and 5.25, respectively. Fig. 5.23 shows for both bands D2 and D2-C. that the collective core angular momentum mainly aligns along the i-axis at I 25h, because the largest moment of inertia is for rotation around the i-axis under the assumption of hydrodynamic moment of inertia. Note that the s-component of the collective core angular momentum is large and cannot be ignored. Moreover, The root mean square components along the intermediate (i-, squares), short (s-, circles) and long (l-, triangles) axes of the rotor, valence protons, and valence neutrons angular momenta calculated as functions of spin by PRM for the doublet bands D2 and D2-C in 136 Nd. it shows a discontinuous behavior between I = 19 and 20h in band D2, and between I = 17 and 18h and I = 22 and 23h in band D2-C, respectively. This is the reason why we observed an abrupt increase of B(M 1)/B(E2) values, as discussed previously. The angular momentum of the three h 11/2 valence proton particles mainly aligns along the s-axis, and those of valence proton and neutron holes mainly along the l-axis. Such orientations form the chiral geometry of aplanar rotation. This is a further strong evidence to support the band D2 and D2-C being chiral bands. Note that due to the large s-component of the rotor and proton, the total angular momentum lies close to the s-i plane. As shown in Fig. 5.23, for the bands D4 and D4-C, the similar aplanar orientation were observed as in band D2. The angular momentum of the rotor mainly aligns along the i-axis, that of the two h 11/2 valence proton particles mainly aligns along the s-axis, that of the two d 5/2 valence protons holes, and that of the two valence neutron holes mainly aligns along the l-axis. Such orientations form a chiral geometry of aplanar rotation. This supports the D4 and D4-C being chiral doublets. A particular feature was observed that at I 19h, the angular momenta of the two h 11/2 valence proton particles tend to align along i-axis. This leads to the increase of B(E2), and hence the decrease of B(M 1)/B(E2) with increasing spin as shown in Fig. 5.22. Note that there is a band-crossing at I = 19h as discussed previously, and the adopted configuration is only suitable for describing the data above bandcrossing.

Discussion

For the chiral doublet bands D5 and D5-C, as shown in Fig. 5.25, the angular momenta have similar orientation at I 21h as in D2. In detail, the angular momentum of the rotor mainly aligns along the i-axis, the two h 11/2 valence proton and one f 7/2 valence neutron particles mainly align along the s-axis, and two g 7/2 valence proton and one neutron h 11/2 valence holes mainly along the l-axis. Compared with those in D2, the s-axis components of the angular momenta of the rotor and h 11/2 valence proton particles in D5 are about 2h smaller. Such orientations form a better chiral geometry of aplanar rotation than that of D2. One observes that at I  21h, the l-axis component of angular momenta of two g 7/2 valence proton holes are different in bands D5 and D5-C. For D5, the two proton holes are aligned and contribute ⇡ 5h. However, for D5-C, the alignment happens when the spin increases from 17h to 21h. At I = 17h, the two proton holes contribute angular momenta ⇡ 2h. At I = 21h, the two proton holes contribute ⇡ 5h. Such difference causes the energy difference between the doublet bands in this spin region ⇡ 400 keV as shown in Fig. 5.20.

From the discussion above, it is clear that the PRM is applied to investigate the five pairs of nearly degenerate doublet bands in the even-even nucleus 136 Nd reproduces well the experimental energy spectra and available B(M 1)/B(E2) values. The angular momentum geometries of the valence nucleons and the core support the chiral rotation interpretation for all the five pairs.

CNS interpretation of all rotational bands

In order to globally understand the structure of all rotational bands in 136 Nd, we performed CNS calculations. The 136 Nd nucleus, with 60 protons and 76 neutrons, can be considered to arise from an interaction between ten valence proton particles above the Z = 50 major shell colsure, and six neutron holes in the N = 82 major shell closure. In the low-energy regime, the nucleus is expected to have a small deformation, " 2 ⇠ 0.15 0.20. Thus, it is convenient to express the single-particle configuration in terms of j shell quantum numbers.

In the CNS formalism the nucleus rotates about one of its principal axes and in the present calculations the pairing correlations are neglected. The deformation is optimized for each single particles configuration explored. The configurations are labelled by the number of particles in low-j and high-j orbitals, respectively, in the different N -shells. The configurations can be defined relative to a 132 Sn core as are not distinguished in the CNS formalism. Note that all particles are listed, i.e. not only the particles considered as active (unpaired). Note also that the labels do not refer to the pure j-shells, but rather to the dominating amplitudes in the Nilsson orbitals. In some cases, for an odd number of particles in a group, the signature will be specified as a subscript + (↵ = +1/2) or -(↵ = 1/2). The A = 130 parameters introduced in Refs. [START_REF] Afanasjev | Termination of rotational bands: disappearance of quantum many-body collectivity[END_REF][START_REF] Bengtsson | Rotational bands and particle-hole excitations at very high spin[END_REF] have been used for the calculation on 136 Nd.

⇡(g) p 1 (dg) p 2 (h 11/2 ) p 3 ⌫(sd) n 1 (h 11/2 ) n 2 (hf ) n 3 (i 13
In the nucleus 136 Nd, the lowest proton configuration has 10 protons in the ⇡g 7/2 and ⇡d 5/2 orbitals which interact and are strongly mixed. Higher angular momenta from proton configurations can be obtained by exciting one, two or three protons from ⇡g 7/2 and ⇡d 5/2 to the ⇡h 11/2 orbitals. The lowest observed bands are characterized by the configurations with the active neutron holes in the ⌫d 3/2 and ⌫s 1/2 orbitals which interact and are strongly mixed. Higher angular momenta from neutron configurations can be generated from neutron configurations with one, two or three active holes in the ⌫h 11/2 orbital instead. Much more excited states and very high angular momenta can be obtained from neutron excitations above the N = 82 shell gap into the ⌫f 7/2 , ⌫h 9/2 and ⌫i 13/2 orbitals and proton excitations from the ⇡g 9/2 orbital across the Z = 50 shell gap.

Before discussing the various observed structures, it is instructive to draw the observed bands relative to a standard rotating liquid drop reference, E E rld (def). The resulting figures reveal not only the relative excitation of the bands, but also details which otherwise are hard to observe in the E I plots. However, the multitude of bands identified in 136 Nd makes their visualisation in a single figure cumbersome. Therefore, we divided them in four groups, which are drawn in different figures of Fig. 5.26, Fig. 5.27, Fig. 5.28, and Fig.

5.29.

As shown in Fig. 5.26, one can observe the up-sloping pattern with increasing spin of all the bands, which is induced by the large difference between the moments of inertia of the low-spin bands and that of the rotating liquid drop. One can also observe the change of slope of the band above spin 10 + , and of the even-spin branch of band N1 above spin 18 + , which are evidently induced by configuration and/or deformation changes.

As shown in Fig. 5.27, one can see that the yrast nature of band L1 in the spin range 10h to 20h, while just above 10 + the lowest excited band is L7. There are three positive-parity yrare bands L2, L3 and L4, two nearly degenerate negative-parity bands L5 and L6 connected to band L1, and two yrare positive-parity bands L8 and L9 which decay to band L7. At spins higher than 20 + there are four bands, labeled T1-T4 to emphasize their interpretation in terms of triaxial bands (see the following sections). Band T2 develops over the largest spin interval, from 15 + to 39 + ; at the highest spins it develops up to ⇡1.7 MeV above yrast, which is a feature similar to that observed in the neighboring 137 Nd [START_REF] Petrache | Collective rotation of an oblate nucleus at very high spin[END_REF]. The bands T1, T3 and T4 have a nearly flat behavior, with moments of inertia which are similar to that of the liquid drop in the observed spin interval.

As shown in Fig. 5.28, one can observe the well known parabolic behavior of the E E rld plots, which is induced by the mismatch between the calculated moment of inertia of the drop and that of a given band: for a perfect matching one would have a flat horizontal line (see e.g. [START_REF] Petrache | Stable triaxiality at the highest spins in 138 Nd and 139 Nd[END_REF]). The chiral doublets are drawn with the same color, but different symbols (circles for the yrast, squares for the yrare bands). The spin of the minimum of each parabola is indicative of the total single-particle spin of the contributing nucleons in the configuration. All bands show nearly degenerate branches with even and odd spins, which indicate the presence in the configurations of one unpaired high-⌦ Nilsson orbital. One can observe that the chiral doublets develop only at high spins for the bands D2, D3, D4, that the energy separation is different for the different doublets, and that the chiral doublet observed over the largest spin interval is that of band D5. with increasing spin of the HD bands, which is induced by the calculations without pairing. The excitation energies of the bands HD2-HD5 are not known experimentally. They are adjusted such that to form two nearly degenerate partners HD1-HD2 and HD3-HD4. This adjustment is in agreement with the yrasness criteria and is supported by the theoretical interpretation presented in the following sections.

The low-and medium-spin bands L and T

The band of 136 Nd exhibits a crossing at spin I ⇡ = 10 + similar to that observed in the neighboring 134 Nd nucleus [START_REF] Petrache | Smooth crossings between theN = 4andN = 6 orbits and adiabatic configuration changes at high angular momentum in 134 Nd[END_REF]. The B(E2;

2 + 2 ! 0 + ) and B(E2; 2 + 2 ! 2 + 1
) values extracted from the relativistic Coulomb excitation measurement reported in Ref. [START_REF] Saito | Yrast and non-yrast 2+ states of 134 Ce and 136 Nd populated in relativistic coulomb excitation[END_REF] clearly show the large triaxiality ( ⇡ 23 ) and pronounced softness of 136 Nd at low spins. The softness is well documented in the A ⇡ 130 mass region, in particular in the 134 Nd nucleus, for which the measured transition probabilities are in good agreement with the O(6) symmetry of the interacting boson model which is adequate for the description of -soft nuclei [START_REF] Klemme | Lifetimes measurements for 134 Nd and neighboring nuclei with the coincidence-plunger technique[END_REF]. Above the crossing at spin I ⇡ = 10 + , the band exhibits a regular increase of the transition energies as expected for a rotational band, but also several transitions towards the bands L1 and L2, which is a clear indication of an important mixing with the configurations of these bands. The same behavior of the bands has been reported in 134 Nd [START_REF] Petrache | Smooth crossings between theN = 4andN = 6 orbits and adiabatic configuration changes at high angular momentum in 134 Nd[END_REF].

As one can observe in Figs. 5.30 and 5.31, the bands L1, L2, L4, and the high-spin part of the band exhibit a similar behavior, are all reasonable well reproduced by the same configuration [START_REF] Akkoyun | Agataadvanced gamma tracking array[END_REF][START_REF] Albrecht | Nuclear physics a. stationary states of nuclei[END_REF], or [⇡h 2 11/2 ] in terms of spherical single-particle orbitals. The calculated deformations in the observed spin range shows their enhanced quadrupole deformation (" 2 ⇡ 0.20) relative to that of the GSB (" 2 ⇡ 0.16), and their pronounced triaxiality ⇡ +25 (see Table 5.5).

The configuration assigned to bands N1 and N2 is [9

+, 1 , 42], or [⇡(dg) 1 h 1 -2 -1 0 1 2 3 E-E rld (def) [MeV]
(a) Exp. 136 Nd

GSB γ-band N1 N2 -2 -1 0 1 2 3 E-E rld (def) [MeV] (b) CNS calculations [ 10 0,42] [82,42] [9 +,-1 -,42]
[10 0,3 -3 +,-] 0 10 20 30

Spin,

I [h -] -2 -1 0 1 2 3 E theo.
-E exp. [ Spin,

I [h -] - 1 0 1 2 3 E theo.
-E exp.

[MeV]

(c) Difference CNS calc. -Exp.

L1- [START_REF] Akkoyun | Agataadvanced gamma tracking array[END_REF][START_REF] Albrecht | Nuclear physics a. stationary states of nuclei[END_REF] L2- [START_REF] Akkoyun | Agataadvanced gamma tracking array[END_REF][START_REF] Albrecht | Nuclear physics a. stationary states of nuclei[END_REF] L3-[9 -1 -, ⌦ ⌫0] in terms of spherical single-particle orbitals, where ⌫0 represents the vacuum for neutrons) and [10 0, 3

+ 3 +, ], or [⇡0 ⌦ ⌫h 1 (sd) 1
] in terms of spherical single-particle orbitals, where ⇡0 represents the vacuum for protons, respectively. The bands have different deformations, which are induced by the different types of active nucleons. As one can observe in Fig. 5.30, a nice global agreement with the experimental bands N1 and N2 is obtained. In addition, band N1 has higher quadrupole deformation (" 2 ⇡ 0.18) than band N2 (" 2 ⇡ 0.16), and positive triaxiality ( ⇡ +25 ) which is opposite to that of band N2 ( ⇡ 25 ). The larger quadrupole deformation and positive triaxiality of band N1 are induced by the low-⌦ h 11/2 proton present in its configuration. The high-⌦ h 11/2 neutron present in the configuration of band N2 induces a smaller increase of the quadrupole defomation and negative triaxiality. Interestingly, the high-spin part of band N1 which exhibits a change of slope in the E E rld plot of Fig. 5.26, is nicely reproduced by the [9 +, 1 , 42] CNS configuration, which shows a jump from the minimum at positive triaxiality (" 2 ⇡ 0.18, ⇡ +25 ) to the minimum at negative triaxiality (" 2 ⇡ 0.17, ⇡ 85 ), indicating a drastic change of the rotation axis, from the intermediate to the long axis, respectively.

The medium-spin bands L5 and L6 are the continuation of the odd-and even-spin cascades composing band N1. The configuration assigned to these bands is [7 +, 3 + , 42] in CNS framework. They have the same neutron configuration as band N1 and two additional h 11/2 aligned protons. In addition, they have pronounced triaxiality, as expected due to the presence of three low-⌦ h 11/2 protons, larger quadrupole deformation (" 2 ⇡ 0.20) than band N1 (" 2 ⇡ 0.18).

The configuration assigned to bands L7, [10 0,24], or ⌫h 2 in terms of spherical orbitals, is the same as proposed in Ref. [START_REF] Paul | Band crossing in the gamma-soft nucleus 136 Nd[END_REF]. It has a smaller deformation (" 2 ⇡ 0.15) than the GSB and negative triaxiality ( ⇡ 35 ), induced by the presence of two high-⌦ h 11/2 neutrons. The continuation of band L7 to higher spins is band L9, to which we assign the [START_REF] Akkoyun | Agataadvanced gamma tracking array[END_REF][START_REF] Peng | Search for multiple chiral doublets in rhodium isotopes[END_REF] configuration involving two h 11/2 aligned protons, which induce a larger quadrupole deformation (" 2 ⇡ 0.18) and positive triaxiality ( ⇡ +27 ). Band L8 has odd spins and is linked to band L7 by weak I = 1 transitions, which most probably have M 1/E2 character. For band L8, a possible configuration [9 1

+ , 3 3 ] is proposed, which is quite different from that of band L7 to which it decays and can explain the weak connecting transitions.

The T bands

In the high-spin region, we observed four bands that are called T bands to underline their pronounced triaxiality, which distinguish them from the other high-spin bands based on nearly axial shapes and are dominated by HD configurations.

Band T1 decays only to the bands L1 and L4, which are well reproduced by the [START_REF] Akkoyun | Agataadvanced gamma tracking array[END_REF][START_REF] Albrecht | Nuclear physics a. stationary states of nuclei[END_REF] configuration involving two h 11/2 aligned protons. As it is not linked through several transitions to the other bands, one reasonably can assume that its configuration is quite different from the configurations of the medium-spin bands. One possible configuration is [7 3 , 3 3 ], which involved one h 11/2 proton and one h 11/2 neutron. One can see in Fig. 5.32, with the configuration [7 3 , 3 3 ], the CNS model can reproduce the band well. Spin, Band T2 is observed over a wide spin range from I = 15 to I = 39, and decays only to band L1. Its increasing excitation energy relative to the other high-spin bands (see Figs. 5.26 and 5.32) is an intriguing behavior that has been recently observed in a band in the neighboring 137 Nd nucleus [START_REF] Petrache | Collective rotation of an oblate nucleus at very high spin[END_REF] and interpreted in terms of rotation of an oblate shape. The suggest possible configuration for band T2 is [82, 3

Discussion

I [h -] -1 0 1 2 3 E theo. -E exp. [MeV] (c) Difference CNS calc. -Exp. T1-[7 -3 -,3 -3 -] T2-[82,3 + 3 + ] T3-[82,3 -3 -] T4-[7 + 3 -,24]
+ 3 + ]
, which involves one more neutron in the h 11/2 orbital. The calculated deformation is (" 2 ⇡ 0.19, ⇡ +26 ) up to spin 19 , while in the range I = 21 to I = 39 the deformation changes gradually from (" 2 ⇡ 0.17, ⇡ 20 ) to(" 2 ⇡ 0.09, ⇡ 54 ). This decreasing quadrupole deformation and increase of the triaxiality with increasing spin is a behavior similar to band O of 137 Nd [START_REF] Petrache | Collective rotation of an oblate nucleus at very high spin[END_REF]. The highly-deformed bands

Discussion

The highly-deformed bands of 136 Nd have been studied relatively long time ago [START_REF] Beck | Superdeformed bands in Nd nuclei[END_REF][START_REF] Clark | A pair of identical superdeformed bands in 136 Nd[END_REF][START_REF] Petrache | Competing mechanism for generating high spin excitations in gamma-soft nuclei: the 136 Nd case[END_REF][START_REF] Perries | Decay-out of the yrast superdeformed band in 136 Nd: Towards an experimental extraction of the neutron pairing gap at large deformation[END_REF]. The decay out of band HD1 has been identified [START_REF] Perries | Decay-out of the yrast superdeformed band in 136 Nd: Towards an experimental extraction of the neutron pairing gap at large deformation[END_REF] and a negative-parity involving two neutrons in the intruder orbitals i 13/2 and (h 9/2 , f 7/2 ) was assigned based on TRS calculations [START_REF] Petrache | Competing mechanism for generating high spin excitations in gamma-soft nuclei: the 136 Nd case[END_REF]. In the present work we identified three more bands developing to very high spins, with behaviors similar to that of the band HD1. Unfortunately, we were not able to identify the deformed entry states and the calculated yrast HD configurations at high spins, we assigned tentative configurations, which compose a quite convincing scenario of the single-particle excitations in the HD well. As can be seen in Fig. 5.34, the yrast configuration with minimum at the right spin and adequate slope for the band HD1 is [64, 5 4(21)], which has positive parity and a close lying [64, 5 + 4(21)] signature partner. One can adjust the excitation energy and spin of band HD2 such that it fits the [64, 5 + 4(21)] configuration. One obtains therefore a new interpretation of the band HD1 of 136 Nd, which now has two neutrons in (h 9/2 , f 7/2 ) and one in i 13/2 , and has positive parity. The situation is a bit more complicated for the configuration assignment to the bands HD3, HD4 and HD5. However, guided by the CNS calcula-Chapter 5. Towards complete spectroscopy of 136 Nd 121 tions and properly adjusting the excitation energies and spins of the bands, while keeping in mind their smaller intensity, we assigned the [64, 45 (21)] and [64, 45 + (21)] configurations to the bands HD3 and HD4, respectively, and the [64, 44( 1+ 1)] configuration to band HD5. With these configuration assignments we obtain a relatively coherent picture of the single-particle excitations in the HD well, which involve always one neutron in the i 13/2 intruder orbital, and one or two neutrons in the (h 9/2 , f 7/2 ) orbitals.

Chapter 6

Evidence of M D in the odd-A nucleus 135 Nd

In this chapter the details of the experimental results and discussion related to the odd-A nucleus 135 Nd are presented. The structure of doublet bands is discussed in the frameworks of the covariant density functional theory and the particle-rotor model.

Introdution

In the A ⇡ 130 mass region, a series of low lying I = 1 chiral bands based on the ⇡h 11/2 ⌦ ⌫h 1 11/2 configuration have been identified in the N = 75 nuclei [START_REF] Starosta | Chiral doublet structures in odd-odd N = 75 isotones: Chiral vibrations[END_REF]. Recently, the multiple chiral doublet (M D) have been reported in the odd-A nucleus 133 Ce [START_REF] Ayangeakaa | Evidence for multiple chiral doublet bands in 133 Ce[END_REF]. M D bands are also expected to exist in the 135 Nd nucleus, which is an isotone of 133 Ce. We therefore performed a careful investigation of the experimental data obtained in our JUROGAM II experiment to possible identify chiral partners of the previously know bands.

Prior to this study, the chirality in 135 Nd has been investigated both experimentally [START_REF] Mukhopadhyay | From chiral vibration to static chirality in 135 Nd[END_REF][START_REF] Zhu | A composite chiral pair of rotational bands in the odd-A nucleus 135 Nd[END_REF]128] and theoretically [51,[START_REF] Zhao | Impact of pairing correlations on the orientation of the nuclear spin[END_REF][START_REF] Brant | Chiral bands in 135 Nd: The interacting boson-fermion model approach[END_REF].

Experimentally, the nearly degenerate doublet bands of negative parity observed in 135 Nd are considered as best examples of chiral bands in the A=130 mass region [START_REF] Zhu | A composite chiral pair of rotational bands in the odd-A nucleus 135 Nd[END_REF]. These bands have been interpreted as chiral doublet bands within the 3D TAC framework. Later, in Ref. [START_REF] Mukhopadhyay | From chiral vibration to static chirality in 135 Nd[END_REF] were reported the lifetime measurement results of intraband and interband transition probabilities of the doublet bands. A microscopic calculation based on the combination of tilted-axis cranking (TAC) model with random phase approximation (RPA) reproduces all experimental observables quite well, substantiating the theoretical interpretation: at the bottom of the bands, the angular momentum vector oscillates perpendicular to the plane spanned by the long and short axes of the triaxial nuclear shape, executing a motion that was called chiral vibration. These oscillations slow down with increasing angular momentum, resulting 123 124 6.2. Experimental results and level scheme in a decreasing energy splitting between the bands and an increase in the interband B(E2) values. The vibration, then, becomes strongly anharmonic, changing into tunneling between well-established left-and right-handed configurations (chiral rotation). This behavior shows how the chirality changes from chiral vibration to nearly static chirality with increasing spin.

A n-particle-n-hole PRM was developed to investigate the chirality in 135 Nd [51] in a fully quantal approach. For the two negative-parity chiral bands, the observed energies and the B(M 1) and B(E2) values for the in-band as well as interband transitions were reproduced excellently. Furthermore, to study in detail the chiral geometry of the aplanar rotation and its evolution with angular momentum, root mean square values of the angular momentum components and their probability distributions were used. The chirality was shown to be a transient phenomenon. The chiral partner bands start as a soft vibration of the angular momentum perpendicular to the plane spanned by the short and long axes, where the band and its partner realized as the zeroand one-phonon state. With increasing angular momentum the vibration becomes strongly anharmonic, progressively localizing in left-and right-handed configurations. Maximal chirality is reached at I = 39/2, where the two bands approach each other closest. At this spin they have very similar distributions of the angular momenta of the core, the valence neutron, and the valence protons in the left-and right-handed sectors, which reflect reduced left right tunneling. With further increasing spin the two bands again develop into the zero and one-phonon states of a chiral vibration of the angular momentum about the intermediate axis [51]. In addition, the doublet bands were also investigated in the framework of interacting boson-fermion model. It has been shown that they can be interpreted as twin chiral bands based on the ⇡h

Experimental results and level scheme

In the present work, we succeeded to identify one new pair of chiral doublet bands in 135 Nd, namely D3 and D4, in addition to the previously know one composed of bands D5 and D6 (see Fig. 6.1), which raises to two the number of chiral doublet bands in this nucleus. In addition, many new transitions have also been observed linking the new bands between them and with the other levels.. The partial level scheme of 135 Nd showing two pairs of doublet bands and their decay toward low-lying states is shown in Fig. 6.1. The experimental information on the observed transitions is given in Table 6.1.

Band D2-3qp, first reported in Ref. [131], is confirmed up to spin 39/2 + . Three new levels are added on top of the band up to spin 45/2 . 6.1), and also on the theoretical interpretation here below, we assign a positive parity to band D2.

Band D3, previously known up to the level de-excited by the 334-keV transition [128], is extended by three more levels up to spin 33/2 + . Another 33/2 + level decaying to both the 31/2 + level of band D3 and to band D2-3qp is also identified. Fourteen new transitions of 377, 389, 420, 566, 589, 612, 616, 666, 679, 705, 832, 917, 1035 and 1069 keV connecting band D3 to band D2-3qp are also identified. The parity of band D3 is changed to positive based on the R DCO and R ac values of the multitude of connecting transitions to band D1 and D2-3qp. In particular, the extracted values for the five connecting transitions of 566, 705, 832, 1035, and 1069 keV clearly indicate their E2 character (see Table . 6.1), therefore firmly establishing the parity of band D3 as identical to the assigned positive parity of band D2 Band D4, previously known up to the state de-excited by the 423-keV transition is confirmed [128], but based on R DCO values of the connecting transitions to bands D1 and D3, the spins are decreased by one unit and the parity is changed to positive. We added one new level with spin 19/2 + at the bottom of the band, one transition of 1237 keV towards band D1, four transitions of 596, 673, 780, and 919 keV towards band D3, and three transitions of 589, 649 and 750 keV towards band D2-3qp. The parity of band D4 is the same as the assigned positive parity of band D3, because the three connecting transitions of 589, 649, and 780 keV between bands 3 and 2 have firmly established E2 character (see Table . 6.1). We therefore adopt a negative parity for band D4. Spectra showing the newly identified transitions in the bands D3 and D4 are given in Fig. 6.2.

We confirm all previously reported levels of bands D5 and D6 in Refs. [START_REF] Mukhopadhyay | From chiral vibration to static chirality in 135 Nd[END_REF][START_REF] Zhu | A composite chiral pair of rotational bands in the odd-A nucleus 135 Nd[END_REF]. Three new levels with spins 21/2 , 45/2 and (47/2 ) are identified at the bottom and at the top of band D5, connected by the new transitions of 130, 560, 565, 1054, and 1125 keV. The two tentative transitions of 557 keV and 963 keV reported previously in Ref. [START_REF] Zhu | A composite chiral pair of rotational bands in the odd-A nucleus 135 Nd[END_REF] are confirmed, but our data show that the energy of the 557-keV transition is instead 559 keV. Three new transitions connecting band D6 to D5 with energies of 609, 997 and 1071 keV are newly identified. Three transitions of 463, 584, and 1299 keV from bands D5 and D6 to band D1 have been also newly identified. The spins and negative parity of band D5 are well established based on the E2 character of the 735-, 835-, and 1299-keV transitions towards band D1, deduced from the measured R DCO and R ac ratios (see Table.

Discussion

To understand the nature of the observed band structure in 135 Nd, we analyzed the excitation energies of the bands, which reveal their detailed structure when drawn relative to a standard rotor reference, like in Fig. 6.3. One can observe a significant signature splitting between the two signatures of the yrast one-quasiparticle band built on the ⌫9/2 [514] Nilsson orbital, which indicates that 135 Nd has a large triaxiality close to the ground state. One can also observe the parabolic behavior of the bands D2-D6, with a difference in excitation energy between the bands D2, D3 and D4 of around 200 keV, and between the bands D5 and D6 of around 500 keV at low spins, which decreases steadily with increasing spin. Band D2 is observed over a much longer spin range than the bands D3, D4. It exhibits a markedly different slope at high spins, indicating a band crossing, and induced us to use the two labels D2-3qp and D2-5qp for the low-and high-spin parts, as they are interpreted as three (3qp) and five-quasiparticle (5qp) bands, respectively (see the discussion below).

Another quantity which reveals the properties of the bands is the single- particle alignment i x , which is given in Fig. 6.4. The used Harris parameters J 0 = 11 h2 MeV 1 and J 1 = 20 h4 MeV 3 which induce a flat behavior of band D2-D4 in the medium spin range, at difference with respect to that of band D1 which is up-sloping with increasing frequency. This clearly shows the polarizing effect of the two additional nucleons contributing to the 3qp configurations of bands D2-D4, which induces a larger deformation. Particular features are exhibited by band D2 at both low and high spins. At low spins one observes a decrease of i x , most probably induced by the interaction with other positiveparity levels not shown in the partial level scheme given in Fig. 6.1. At high spins one observes a backbending with a spin gain of at least 7h, indicating the alignment of two more nucleons, most probably protons occupying the h 11/2 orbital. Furthermore, the very similar values of around 9h exhibited by the bands D2-D4, suggest similar 3qp configurations. The difference of ⇡ 7h at low frequency between the bands D2-D4 and D1, strongly suggests the involvement of two more nucleons placed on opposite parity orbitals in the bands D2-D4, most probably protons occupying the h 11/2 and the strongly mixed (d 5/2 , g 7/2 ) orbitals. The single-particle alignment i x of the bands D5 is larger by 2-3h than that of the bands D2-D4, while that of band D6 is larger than that of band D5 by ⇡ 2h. The higher spin alignment of the bands D5, D6 relative to that of bands D2-D4 clearly shows the involvement of a pair of protons in the h 11/2 orbital.

A third quantity revealing the properties of the band is the ratio of reduced transition probabilities B(M 1)/B(E2), which are given together with the calculated values in Fig. 6 In order to examine in detail the chiral character of the observed bands, the recently developed particle-rotor model has been used [51,55]. The input deformation parameters ( 2 , ) for PRM calculations are obtained from constrained CDFT calculations [START_REF] Meng | Possible existence of multiple chiral doublets in 106 Rh[END_REF]. In addition, for the electromagnetic transitions, the empirical intrinsic quadrupole moment

Q 0 = (3/ p 5⇡)R 2 0 Z with R 0 = 1.2A 1/
3 fm, and gyromagnetic ratios for rotor g R = Z/A and for nucleons g p(n) = g l + (g s g l )/(2l + 1) [g l = 1(0) for protons (neutrons) and g s = 0.6g free ] [3] are adopted.

The ground state band D1 with the configuration ⌫(1h 11/2 ) 1 has been studied previously [START_REF] Zhu | A composite chiral pair of rotational bands in the odd-A nucleus 135 Nd[END_REF]131,[START_REF] Beck | Superdeformed band in 135 Nd[END_REF]. The deformation parameters obtained from the CDFT calculations are 2 = 0.19 and = 25.5 , which are similar to those used in Ref. [133], in which the measured transition probabilities were well reproduced. The obtained energy spectra, which are shown in Fig. 6.3, are in excellent agreement with the data in the low spin region. The deviation from the experimental data at high spin is due to the interaction with band D5 present work are very similar to those obtained from the results published in Ref. [START_REF] Mukhopadhyay | From chiral vibration to static chirality in 135 Nd[END_REF]. The calculated and experimental B(M 1)/B(E2) ratios are showed in Fig. 6.5 (a). One can see that the PRM values are in good agreement with the experimental data, including the sudden increase occurring at spin 39/2, which is induced by a sudden decrease of the B(E2) values at spins above 39/2, and was interpreted as due to the transition from chiral vibration to chiral rotation.

Discussion

The success in reproducing the excitation energies and the electromagnetic transition probabilities of the bands D5 and D6 by the PRM calculations give a strong support to the configuration assignment of ⇡(1h 11/2 ) 2 ⌦ ⌫(1h 11/2 ) 1 . In order to investigate to what extent the 3D chiral geometry is present in the two chiral doublets, we plotted the components of the angular momenta on the three axes (short, intermediate and long) of the intrinsic reference system. The results for bands D3 and D4 are shown in Fig. 6.6. Those for bands D5 and D6 are similar to those reported in Ref. [START_REF] Mukhopadhyay | From chiral vibration to static chirality in 135 Nd[END_REF], hence here we do not present them once again. One can observe a significative difference between the two chiral doublets: the positive-parity configuration assigned to bands D3 and D4 fulfils much better the chiral geometry, exhibiting equilibrated single-particle angular momenta along the three axes. The negative-parity configuration assigned to band D5 and D6 exhibits a higher single-particle angular momentum along the short axis, which brings the total angular momentum closer to the short-long principal plane and facilitates the chiral vibration at low spin. 135 Nd. The observed doublet bands are compared with CDFT and PRM calculations, which nicely reproduce the experimental data. According to these results, the existence of the M D phenomenon in nuclei of the A ⇡ 130 mass region is strongly supported. The present results encourage us to continue the study of chirality in nuclei, both experimentally (measurement of lifetimes and search for chiral doublets in other nuclei) and theoretically. Table 6.1 -Experimental information including the -ray energies, energies of the initial levels E i , intensities I , anisotropies R DCO and or R ac , multipolarities, and the spin-parity assignments to the observed states in 135 Nd. The transitions listed with increasing energy are grouped in bands and the transitions connecting a given band to low-lying states are listed at the end of each band separated by a blank line. The deduced values for R DCO with a stretched quadrupole gate are ⇡ 1 for stretched quadrupole and ⇡ 0.46 for dipole transitions, while the ratio is close to 1 for a dipole and 2.1 for a quadrupole transition when the gate is set on a dipole transition. The R ac values for stretched dipole and quadrupole transitions are ⇡ 0.8 and ⇡ 1.4. Chapter 7

E a E i (keV) I b R DCO c R ac d Mult. J ⇡ i ! J ⇡ f Band D1 198
E i (keV) I b R DCO c R ac d Mult. J ⇡ i ! J ⇡ 6.3. Discussion Table 6.1 -Continued E a E i (keV) I b R DCO c R ac d Mult. J ⇡ i ! J ⇡ f Table 6.1 -Continued E a E i (keV) I b R DCO c R ac d Mult. J ⇡ i ! J ⇡ f 6.3. Discussion c R

Search for long-lived isomeric states

In this chapter, the recoil-decay tagging technique will be briefly introduced. Then, the experimental results obtained from the analysis of the data collected at the focal plane will be presented.

Introduction

The nuclei around the N = 82 shell closure can have a variety of shapes, such as spherical at low spins, triaxial, highly deformed and superdeformed at high spins. At difference with respect to the well deformed nuclei which can exhibit long and regular rotational bands up to very high spins, the decay flux in the nearly spherical nuclei is often fragmented and interrupted by isomeric states, which renders more difficult their observation. Such a situation is encountered in the weakly deformed Nd nuclei, i.e., 139 Nd (I ⇡ = 23/2 + , T 1/2 = 272 ns) and 140 Nd (I ⇡ = 7 , T 1/2 = 600 µs) [START_REF]ENSDF database[END_REF], with neutron numbers close to N = 82 shell closure, which exhibit irregular sequences of transitions and possible yrast traps. Moreover, isomeric states were also observed in the well deformed Nd nuclei, i.e., 134 Nd (I ⇡ = 8 , T 1/2 = 410 µs), 137 Nd (I ⇡ = 11/2 , T 1/2 = 1.6 s) [START_REF]ENSDF database[END_REF]. This encouraged us to search for possible longlived isomeric states in the present experimental populated nuclei, in particular in 135 Nd and 136 Nd nuclei.

The recoil-decay tagging technique

The recoil-decay tagging (RDT) [START_REF] Schmidt | Gamma-spectroscopic investigations in the radiative fusion reaction 90 Zr + 90 Zr[END_REF][START_REF] Simon | Evidence for nuclear shape coexistence in 180 Hg[END_REF][START_REF] Paul | In-beam -ray spectroscopy above 100 Sn using the new technique of recoil decay tagging[END_REF] is based on the use of temporal and spatial correlations of events in the measurement. The JUROGAM II + RITU + GREAT setup offers such a possibility. The fusion-evaporation reaction produces excited residual nuclei at the target position where they emit prompt rays in the de-excitation process to the ground state. These 139 140 7.3. Results of the focal plane prompt events are detected by JUROGAM II array. The reaction recoils then flight through the RITU separator are implanted into the focal plane GREAT array where one can measure delayed events resulting from long-lived isomers, with lifetimes sufficiently long to allow their observation after the flight time through the spectrometer. To correlate the events recorded by the detectors placed at different locations, the TDR data acquisition system was always used. In the RDT method, the "tag" can be charged particles, i.e., proton, ↵ and .

In the prompt and delayed spectroscopy, to correlate the prompt and delayed ray transitions across isomeric states and identify the lifetime of the isomer, the RDT technique can be applied. In addition, the rays following deexcitation of an isomeric state and detected at the focal plane of a recoil separator can be used to identify prompt ray transitions feeding the isomeric state. This method have been turned out to well suit for identifying the isomers with a half-live larger than the flight time through RITU [START_REF] Paul | In-beam -ray spectroscopy above 100 Sn using the new technique of recoil decay tagging[END_REF].

Results of the focal plane

In the present work, initially, for the identification of the the rays of a specific nucleus we produced -matrices between the delayed transition detected by the clovers placed at the focal plane. In addition, to search for the transitions feeding a long-lived isomer we also produced -matrices between the JUROGAM II array (detecting prompt rays) and the clovers detectors (detecting delayed rays) placed at the focal plane. Various time conditions can be applied to these matrices to optimize the sensitivity for the isomeric decay with different half-lives. These matrices allowed the study of correlated prompt and delayed rays across isomeric states. Note that the time of theray emission measured at the focal plane is relative to the time of implantation of the recoiling nucleus in the DSSD detector.

After a careful analysis of the matrix obtained from the focal plane Ge detectors, we can select the cleanest transitions to extract the prompt-delayed spectra between JUROGAM II and the detectors at the focal plane. Thus, we can link the "new" transitions to the known level scheme of nucleus in this step.

To identify and extract the lifetime of the isomeric states, matrices can be constructed of delayed ray energy versus the time of the focal plane clovers ( see Fig. 7.1). The time defined by a start induced by a recoil event in the DSSD detector and a stopped induced by a delayed ray detected in the focal plane clover detectors. The half-life can be obtained by gating on the rays de-exciting the isomeric states of interest and then fitting the time spectra. By constructing several -Time matrices with various constraints on the time of the delayed ray, we also observed and confirmed the 11/2 , T 1/2 = 2.7 µs isomer of 137 Pr and the 6 + , T 1/2 = 90 ns isomer of 136 Pr, but we could not extract lifetimes more precise than those already known [START_REF]ENSDF database[END_REF]. However, no long-lived isomeric states have been found in 135 Nd and 136 Nd in the present data. In order to show the details of application of the RDT method in this work, we discuss here the already known 10 + , T 1/2 = 370 ns isomer of 138 Nd as an example. A partial level scheme of 138 Nd relate to the isomeric state is given in Fig. 7.2.

In the first step, we identified the 521-, 729-, 884-and 973-keV transitions of ground state band in 138 Nd from the -matrices of the delayed rays detected by the clovers placed at the focal plane.

In the second step, the clean 521-, 729-, 884-and 973 transitions were selected to extract the prompt-delayed spectra between JUROGAM II and the detectors at the focal plane. An example of a prompt spectrum from JU-ROGAM II obtained by gating on selected transitions of 138 Nd measured at the focal plane is shown in Fig. 7.3. One can see the 193-, 231-, 286-, 381-and 647 keV transitions in this spectrum, which belong to the prompt transitions above the isomer of the 138 Nd. Finally, we can extract the lifetime from time spectra of rays in the deexcitation cascade below the isomeric state, after proper background subtraction and fit with an exponential decay curve. In the nucleus 138 Nd, the lifetime of 10 + , T 1/2 = 370 ns isomer is deduced from sum spectra of the 521,729, 884, and 973 keV transitions below the isomeric state. Time spectra of the transitions deexciting the isomeric states and a representative fit are shown in Fig. 7.4. The half-life of the 10 + isomer is 374 ± 15 ns in present work could not be more precise than previously reported result 370 ± 5 [START_REF]ENSDF database[END_REF].

Synthèse Introduction

Le terme chiralité a été introduit pour la première fois par Lord Kelvin en 1904 dans ses conférences à Baltimore: « J'appelle n'importe quelle figure géométrique, ou groupe de points, chirale, et dis qu'elle a la chiralité, si son image dans un miroir plan, idéalement réalisée, ne peut être amenée à coïncider avec elle-même. » La chiralité existe généralement dans la nature et a des conséquences importantes dans des domaines scientifiques aussi divers que la biologie, la chimie et la physique. Les exemples les plus connus d'objets géométriquement chiraux sont les mains humaines et le caractère microscopique de certaines molécules. La symétrie chirale est également bien connue en physique des particules, où elle est de nature dynamique et distingue les deux orientations possibles du spin intrinsèque par rapport à la quantité de mouvement de la particule. En physique nucléaire, la chiralité a été suggérée en 1997 par Frauendorf et Meng. Il apparaît dans un noyau triaxial qui tourne autour d'un axe situé hors des trois plans principaux de la forme nucléaire ellipsoïdale.

Des approches adiabatiques et à champ moyen relativiste triaxial contraint à configuration fixée ont été développées pour étudier la coexistence de formes triaxiales et les bandes de doublets chirales possibles en 2006, qui prédit un nouveau phénomène, à savoir l'existence de multiples doublets chiraux (MχD), c'est-à-dire plusieurs paires de doublet chiraux dans un seul noyau. Ce phénomène a été suggéré pour 106 Rh après avoir examiné l'existence possible d'une déformation triaxiale et de la configuration correspondante de trous de proton et de particules de neutron à j élevé. Cette étude a été étendue aux autres isotopes du rhodium et l'existence de MχD a été suggérée dans les noyaux Rh. L'étude a prédit MχD non seulement dans 106 Rh, mais également dans d'autres régions de masse, c'est-à-dire A ≈ 80 et A ≈ 130.

Il convient de noter que, jusqu'à présent, toutes les bandes MχD observées ne sont que dans des noyaux impair-impair, impair-pair et pair-impair. Cela nous a inspiré à rechercher les bandes MχD dans les noyaux pairs. C'est l'un des objectifs principaux de l'expérience présentée dans cette thèse. D'un point de vue expérimental, les bandes de doublets doivent répondre à un ensemble de critères pour être reconnues en tant que bandes chirales, parmi lesquelles les plus importantes sont la séparation de l'énergie entre les partenaires, leurs taux de transitions électromagnétiques et les géométries de moment angulaire des bandes doublet observées, investigués en calculant les valeurs moyennes des composantes en moment angulaire du noyau, des protons de valence et des neutrons de valence, le long des axes intermédiaire, court et long. L'essentiel du travail actuel a été entrepris dans le but de rechercher plusieurs bandes chirales dans le 136 Nd, un noyau qui est le voisin le plus proche du premier candidat chiral 134 Pr et du premier wobbler 135 Pr rapporté en dehors de la région A ≈ 160. De plus, un effort important a été consacré à la spectroscopie détaillée du 136 Nd, car la plupart des résultats expérimentaux rapportés précédemment ont été obtenus il y a plus de vingt ans avec des matrices de détecteurs moins efficaces.

L'autre objectif du présent travail de thèse était axé sur une étude détaillée de la chiralité dans 135 Nd, afin de rechercher plusieurs bandes chirales similaires à celles observées dans l'isotone du 133 Ce. Le noyau 135 Nd est l'un des exemples les plus connus de vibrations chirales.

Outre l'étude des bandes chirales de 135 Nd et 136 Nd, nous avons consacré un effort important à la recherche d'isomères à longue durée de vie dans les noyaux peuplés, en particulier dans 135 Nd et 136 Nd, en utilisant la configuration JURIGAM II + RITU + GREAT et la technique de marquage par noyaux de recul.

Détails expérimentaux et traitement des données

L'expérience a été réalisée à l'Université de Jyväskylä, en Finlande, avec l'installation JUROGAM II + RITU + GREAT. Des états de spin élevé dans 135,136 Nd ont été peuplés en utilisant la réaction de fusion-évaporation 100 Mo ( 40 Ar, xn). Le faisceau d'ions 40 Ar a été produit dans une source d'ions ECR et accéléré par le cyclotron K130 jusqu'à une énergie finale de 152 MeV. L'intensité du faisceau était d'environ 14 pnA. Une semaine de temps de faisceau a été allouée à cette expérience. Nous avons utilisé comme cible une feuille autoportante enrichie de 100 Mo d'une épaisseur de 0.5 mg/cm 2 . Les noyaux 135 Nd et 136 Nd étaient les plus fortement peuplés dans la réaction, avec des sections efficaces d'environ 100 mb chacune, pour une section efficace totale de la réaction calculée avec PACE4 de 480 mb.

La boule de détecterus Germanium JUROGAM II, un spectromètre composé de 24 détecteurs trèfle et 15 détecteurs coaxiaux tronconiques placés à la position de la cible, a été utilisé pour détecter des rayons gamma promptes. Les détecteurs de trèfle ont été placés sur deux anneaux à 75.5º (12 détecteurs) et 104.5º (12 détecteurs) symétriques par rapport à 90º. Les détecteurs coniques ont également été placés sur deux anneaux à 133.6º (10 détecteurs) et 157.6º (5 détecteurs). Tous les détecteurs Ge sont équipés d'écrans anti-Compton constitués de collimateurs BGO et Hevimet placés devant les détecteurs afin d'améliorer le rapport P/T. La méthode d'addition consistant à additionner les énergies de rayons gamma coïncidentes dans des cristaux séparés dans le même détecteur de trèfle dans une fenêtre temporelle de 200 ns a été utilisée dans ce travail. Pour un rayon gamma avec une énergie de 1.33 MeV, le spectromètre complet JUROGAM avec 39 détecteurs a une efficacité de photopic totale d'environ 5% et une résolution en énergie de 3.5 keV (FWHM).

Les séparateurs de recul pour les produits de fusion induits par les ions lourds sont devenus des outils importants pour séparer les résidus d'évaporation produits par la fusion du faisceau primaire avec la cible des d'autres produits de réaction indésirables, puis pour transporter les reculs sur le plan focal en vue de mesures ultérieures (comme par exemple les propriétés de désintégration de éléments lourds, recherche d'états isomériques à longue durée de vie dans les noyaux). L'unité de transport des ions de recul (RITU) est un séparateur de recul rempli de gaz, qui est couplé au multidétecteur JUROGAM II. Il se compose d'un total de quatre éléments focalisants qui sont dans une configuration Q 1 DQ 2 Q 3 .

Le spectromètre GREAT (Gamma REcoil Alpha Tagging) est placé dans le plan focal du séparateur RITU. Il est conçu pour mesurer les propriétés de désintégration des produits de réaction transportés dans le plan focal du séparateur à recul. En outre, GREAT peut être utilisé en tant que dispositif autonome sensible pour les mesures de désintégration dans le plan focal ou pour fournir une étiquette efficace permettant une conversion rapide des électrons ou des rayons mesurés avec des réseaux de détecteurs déployés à la position cible. Le réseau de détecteurs GREAT est composé de plusieurs types de détecteurs: compteur proportionnel multifils (MWPC), détecteurs segmentés de silicium double face (DSSD), diodes PIN, un détecteur planaire et trois détecteurs trèfle en Germanium.

Dans cette expérience, ≈ 4000 Go de données réparties sur 2000 fichiers ont été stockées. Un total de 5.1×10 1 0 événements de coïncidence des rayons gamma avec multiplicité ≥ 3 ont été recueillis. Toutes les données ont été enregistrées par le système d'acquisition de données TDR sans déclenchement et les événements ont été horodatés en utilisant une horloge de 100 MHz avec une résolution temporelle de 10 ns. Dans mon analyse de données, j'ai effectué un étalonnage en énergie, une adaptation du gain et étalonnages d'efficacité pour les détecteurs Ge, correction de déplacement Doppler, addition de des signaux pour les détecteurs de trèfle. Afin d'attribuer ls spin et la parité pour les nouveaux niveaux identifiés, les multipolarités des rayons gamma ont été extraites à l'aide des rapports DCO (CorrélationDdirectionnelle à états Orientés) et des rapports de corrélation angulaire à deux points (Anisotropie) R ac .

Dans ce travail, les valeurs de R DCO ont été extraites de matrices gamma-gamma qui ont été formées en triant les événements de coïncidence rapides avec les détecteurs à (157.6º) par rapport à ceux à (75.5º et 104.5º) correspondant à l'angle autour de 90º.

Dans la présente analyse des données, les valeurs de R ac ont été extraites de matrices gammagamma, qui ont été formées en triant les événements de coïncidence rapides avec des combinaisons (133.6º et 157.6º) versus (tous les angles) et (75.5º et 104.5º) versus (tous les angles), en définissant les mêmes fenêtres d'énergie sur le spectre de projection (tous les angles) dans les deux matrices et en projetant sur l'autre axe. Ensuite, le ratio de R ac a été calculé en utilisant les intensités extraites des rayons gamma d'intérêt de ces spectres, normalisées par l'efficacité différente des deux ensembles de détecteurs.

Les valeurs déduites pour R DCO avec une fenêtre sur une transition quadripolaire étirée sont 1 pour un quadripôle étiré et 0.46 pour des transitions dipolaires, tandis que le rapport est proche de 1 pour un dipôle et de 2.1 pour une transition quadripolaire lorsque la fenêtre est posée sur une transition dipolaire. Les valeurs de R ac pour les transitions dipolaires et quadripolaires étirées sont 0.8 et 1.4.

Résultats expérimentaux

Dans notre travail, le schéma de niveaux de 136 Nd a été considérablement étendu. De nombreuses nouvelles bandes ont été identifiées à la fois à bas et à haut spin, parmi lesquelles cinq bandes presque dégénérées interprétées comme des partenaires chiraux. En outre, les énergies d'excitation, les spins et les parités de plusieurs bandes connues auparavant sont révisés et fermement établis. La plupart des transitions observées ont été regroupées en bandes: la bande γ, deux bandes de parité négative à bas spins (N1,N2), neuf bandes à spin moyen (L1-L9), six bandes de transitions dipolaires (D1-D6) et cinq bandes partenaires chirales (D1-chirale, D2-chirale, D3-chirale, D4chirale, D5-chirale), quatre bandes à spin élevé (T1-T4) et cinq bandes fortement déformées (HD1-HD5).

La bande γ, précédemment connue jusqu'au spin I = 5 + , est maintenant étendue à des spins plus élevés et liée par les transitions de 815 et 600 keV à une autre bande précédemment connue, construite sur l'état 10 + à 3768 keV. La bande γ est maintenant observée jusqu'à I = 20 + . Pour la bande N1, nous avons récemment identifié les transitions de 1077, 410, 1079, 744 et 832 keV placées au dessus des états précédemment connus. De plus, nous avons observé deux nouvelles transitions de 381 et 612 keV de l'état 10 + du GSB vers l'état intermédiaire 9 -, qui à leur tour se désintégraient vers les états 7 -et 8 -de la bande N1, et directement vers l'état 9 -état de la bande N1, respectivement. La bande N2 est observée pour la première fois. Nous avons beaucoup étendu les bandes L1-L9, dans lesquelles de nombreux nouveaux états ont été observés. Une caractéristique particulière du schéma de niveau de 136 Nd est l'existence de cinq paires de bandes presque dégénérées (D1 et D1-chirale, D2 et D2-chirale, D3 et D3-chirale, D4 et D4-chirale, D5 et D5-chirale) et la bande dipôlaire D6 sans partenaire. Nous avons récemment identifié les partenaires chiral des bandes D1-D5. Cinq bandes hautement déformés (HD) ont été observés dans le 136 Nd dans le présent travail. Les trois bandes HD2, HD4 et HD5 sont nouvelles. Toutes les bandes coïncident avec les transitions connues de 136 Nd. Cependant, aucune transition de liaison vers les états normaux déformés n'a été trouvée. Les spins, les parités et les énergies d'excitation ont été attribués sur la base des calculs du modèle CNS.

De plus, nous avons réussi à identifier une nouvelle paire de bandes chirales dans 135 Nd, à savoir D3 et D4, en plus de la précédente, composée des bandes D5 et D6, ce qui porte à deux le nombre doublets chirales dans ce noyau. En outre, de nombreuses nouvelles transitions ont également été observées reliant les nouvelles bandes entre elles et avec les autres niveaux. La bande D2-3qp, connue précédemment, est confirmée jusqu'au spin 39/2 + . Trois nouveaux niveaux sont ajoutés au sommet de la bande jusqu'à spin 45/2 + , qui, avec le niveau précédemment connu 39/2 + , forment une nouvelle bande nommée D2-5qp. La bande D3, jusque-là connue jusqu'au niveau désexcité par la transition à 334 keV, est étendue de trois niveaux supplémentaires jusqu'au spin 33/2 + . Un autre niveau de 33/2 + qui se désexcite à la fois vers le niveau 31/2 +

Conclusion et perspectives

La présente thèse présente l'étude spectroscopique des noyaux triaxiaux 135,136 Nd, avec un accent particulier sur le phénomène des doublets chiraux multiples (MχD). Un effort important a également été consacré à la recherche d'isomères à longue durée de vie possibles dans les 135 Nd et 136 Nd, en utilisant la configuration JUROGAM II + RITU + GREAT et la technique de marquage par décroissance de recul.

Dans 136 Nd, la caractéristique la plus importante du schéma de niveau est l'observation des cinq paires de bandes rotationnelles presque dégénérées, qui ont des propriétés en accord avec un caractère chiral. Les bandes observées ont été étudiées par la théorie fonctionnelle de densité covariante avec rotation contrainte autour d'un axe incliné (TAC-CDFT). Les configurations possibles ont été explorées. Les spectres d'énergie expérimentaux, les moments cinétiques et les valeurs B(M1)/B (E2) des configurations attribuées sont globalement bien reproduits par les calculs TAC-CDFT. Les résultats théoriques soutiennent l'interprétation chirale des bandes observées, qui correspondent à des formes avec une triaxialité presque maximale induite par différentes configurations multi-quasi-particules dans 136 Nd. Plus tard, inspiré par nos résultats expérimentaux, Qibo Chen et al. ont mis au point un modèle particule-rotor (PRM) couplant les nucléons de quatre couches j à un rotor triaxial. Il a été constaté que les propriétés de ces bandes doublet sont en bon accord avec les résultats des calculs de PRM. Par conséquent, le phénomène MχD dans le noyau 136 Nd a été confirmé. A noter que c'est pour la première fois que des bandes MχD sont observées dans un noyau à des spins élevés. De plus, les cinq paires de bandes chirales observées dans 136 Nd est le plus grand nombre de bandes chirales observées dans un seul noyau jusqu'à présent. Tout récemment, un nouveau travail théorique inspiré pas nos résultats a également été consacré à l'étude des cinq bandes chirales de 136 Nd, dans lequel le modèle en couche projeté triaxial incluant des configurations avec plus de quatre quasi-particules dans l'espace de configuration a été développé.

En outre, le schéma de niveau de 136 Nd a été considérablement étendu à des spins bas, moyens et très élevés. Beaucoup de nouveaux états et de nouveaux bandes ont été identifiés dans ce noyau. Les énergies d'excitation, les spins et les parités des bandes précédemment connues ont été révisés et fermement établis. Des configurations ont été attribuées aux bandes observées sur la base de calculs Nilsson-Strutinsky en rotation contrainte. La structure de bande de 136 Nd a été clarifiée et les différents types d'excitations uni-particule et collectives ont été bien compris.

Dans 135 Nd, une nouvelle paire de bandes chirales à parité positive a été identifiée qui, avec les bandes chirales à parité négative précédemment rapportées, constitue un troisième cas de bandes MχD dans la région de masse A ≈ 130. Ces bandes chirales comprennent des transitions dipolaires et sont caractérisées par la même parité, des énergies d'excitation des niveaux très proches et une variation régulière du paramètre S(I). Les caractéristiques des bandes doublet soutiennent l'interprétation chirale. De plus, les bandes doublet observées ont été comparées à la théorie fonctionnelle de densité covariante sous contrainte (CDFT) et aux calculs PRM reproduisant fidèlement les données expérimentales, confirmant le phénomène MχD dans ce noyau. Les bandes MχD nouvellement observées dans 135 Nd représentent une fort support de l'existence de MχD dans les noyaux.

La recherche d'états isomériques à longue durée de vie dans 135 Nd et 136 Nd n'a pas conduit à l'identification de nouveaux isomères, mais nous avons pu confirmer les isomères rapportés dans les noyaux 138 Nd, 134 Ce, 136 Pr et 137 Pr dans nos données.

Selon ces résultats, l'existence du phénomène MχD dans les noyaux de la région de masse A ≈ 130 est fortement confirmée. Les présents résultats nous incitent à poursuivre l'étude de la chiralité dans les noyaux, à la fois expérimentalement (mesure de la durée de vie et la recherche de doublets chiraux dans d'autres noyaux) et théoriquement.

Conclusions and Outlook

The present thesis presents the spectroscopic study of the triaxial nuclei 135,136 Nd, with particular emphasise on the multiple chiral doublets (M D) phenomenon. An important effort was also devoted to search for possible longlived isomers in 135 Nd and 136 Nd, using the RITU+GREAT setup and the recoil decay tagging technique.

The high-spin states in the nuclei 135,136 Nd were populated using the the fusion-evaporation reaction 100 Mo( 40 Ar,xn) at a beam energy of 152 MeV, provided by the K130 Cyclotron at the University of Jyväskylä, Finland. The high efficiency JUROGAM II array was used to detect prompt rays, resulting in a high statistics data set which allowed us to extend the level scheme of 135,136 Nd significantly. The RITU spectrometer and its focal plane array GREAT were used to measure delayed events, allowing us to search for long-lived isomers in the populated nuclei.

In 136 Nd, the most important feature of the level scheme is the observation of the five pairs of nearly degenerate rotational bands, which have properties in agreement with a chiral character. The observed bands were investigated by the constrained and tilted axis cranking covariant density functional theory (TAC-CDFT). Possible configurations have been explored. The experimental energy spectra, angular momenta, and B(M 1)/B(E2) values for the assigned configurations are globally well reproduced by the TAC-CDFT calculations. The theoretical results support the chiral interpretation of the observed bands, which correspond to shapes with nearly maximum triaxiality induced by different multiquasiparticle configurations in 136 Nd. Later, inspired by our experimental results, a particle-rotor model (PRM), which couples nucleons in four single-j shells to a triaxial rotor core was developed by QiBo Chen et al [55], to investigate the five pairs of nearly degenerate doublet bands of 136 Nd. It was found that the properties of these doublet bands are in good agreement with results of the PRM calculations. Therefore, the M D phenomenon in the nucleus 136 Nd was further confirmed. Note that it was the first time that M D bands were observed in an even-even nucleus at high spins, and also five pairs of chiral doublet bands is the largest observed in a single nucleus until now. Very recently, a new theoretical work was also devoted to the study of the five chiral doublet bands of 136 Nd [START_REF] Wang | Multichiral facets in symmetry restored states: Five chiral doublet candidates in the even-even nucleus 136 Nd[END_REF], in which the triaxial projected shell model including configurations with more than four quasiparticles in the 147 148 7.3. Results of the focal plane configuration space was developed.

In addition, the level scheme of the 136 Nd has been extended considerably at low, medium, and very high spins. Many new states and new bands have been identified in this nucleus. Excitation energies, spins, and parities of the previously known bands have been revised and firmly established. Configurations have been assigned to the observed bands based on cranked Nilsson-Strutinsky calculations. The band structure of 136 Nd was clarified and the various types of single-particle and collective excitations were well understood.

In 135 Nd, one new pair of positive-parity chiral doublet bands have been identified which together with the previously reported negative-parity chiral doublet bands constitute a third case of M D bands in the A ⇡ 130 mass region. The characteristics of the doublet bands support the chiral interpretation. Furthermore, the observed doublet bands were compared with constrained covariant density functional theory (CDFT) and PRM calculations which nicely reproduce the experimental data, confirming the M D phenomenon in this nucleus. The newly observed M D bands in 135 Nd represents an important milestone in supporting the existence of M D in nuclei.

The search for long-lived isomeric states in 135 Nd and 136 Nd did not lead to the identification of new isomers, but we could confirm the reported isomers in the nuclei 138 Nd, 134 Ce, 136 Pr, and 137 Pr in our data.

According to these results, the existence of the M D phenomenon in nuclei of the A ⇡ 130 mass region is strongly supported. The present results encourage us to continue the study of chirality in nuclei, both experimentally (measurement of lifetimes and search for chiral doublets in other nuclei) and theoretically. La recherche d' états isom ériques à longue dur ée de vie dans 135 Nd et 136 Nd a ét é aussi effectu ée, mais n'a pas permis d'identifier de nouveaux isom ères. Cependant, nous avons pu confirmer la pr ésence des isom ères d éj à identifi és dans les noyaux 138 Nd, 134 Ce, 136 Pr, et 137 Pr dans nos donn ées. Title : Chirality in the 136 Nd and 135 Nd nuclei Keywords : Nuclear spectroscopy, high-spin states, isomers, triaxiality, chirality Abstract : The exotic collective excitation mode called chirality has been investigated and long-lived isomers have been searched for in the 136 Nd and 135 Nd nuclei. Five pairs of nearly degenerate I=1 bands with the same parity have been identified at high spins in 136 Nd. The observed bands were investigated by the constrained and tilted axis cranking covariant density functional theory and a new developed four singlej shells particle-rotor model. It was found that the properties of these doublet bands are in good agreement with results of the model calculations. Therefore, the multiple chiral doublets (M D) phenomenon in the nucleus 136 Nd was confirmed. This was the first experimental evidence for the M D bands in even-even nuclei. The five pairs of chiral doublet bands is the largest observed in a single nucleus until now. In addition, the level scheme of the 136 Nd has been extended significantly at low, medium, and very high spins. Possible configurations of all rotational bands have been assigned by using the cranked Nilsson-Strutinsky model. The band structure of 136 Nd was clarified and the various types of single-particle and collective excitations were well understood.

A new pair of positive-parity chiral doublet bands has been identified in 135 Nd. The characteristics of the doublet bands support the chiral interpretation. Together with the previously reported negative-parity chiral doublet bands show the presence of M D bands in 135 Nd. The observed doublet bands were compared with constrained covariant density functional theory and particle-rotor model calculations which nicely reproduce the experimental data, confirming the M D phenomenon in this nucleus. The newly observed M D bands in 135 Nd represents an important milestone in supporting the existence of M D in nuclei.

The search for long-lived isomeric states in 135 Nd and 136 Nd has also been performed. This did not lead to the identification of new isomers, but we could confirm the existence of the previously reported isomers in the nuclei 138 Nd, 134 Ce, 136 Pr, and 137 Pr in our data.

Universit é Paris-Saclay

Espace Technologique / Immeuble Discovery Route de l Órme aux Merisiers RD 128 / 91190 Saint-Aubin, France

36 2 . 8 69 5. 1

 28691 Schematic diagram of the strong coupling limit in the particlerotor model. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39 2.9 Schematic diagram of the decoupling limit in the particle-rotor model. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39 2.10 Discrete symmetries of the mean field of a rotating triaxial reflection symmetric nucleus. The axis of rotation (z) is marked by the circular arrow. The rotational band structures associated with each symmetry type are presented on the right side. This figure was taken from Ref. [7]. . . . . . . . . . . . . . . . . . . 49 3.1 Schematic illustration the heavy-ion fusion-evaporation reaction forming a compound nucleus and its decay. . . . . . . . . . . . 54 3.2 (a) Photoelectric absorption, (b) Compton scattering, (c) Pair production. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 55 3.3 Reverse bias of the p-n junction. . . . . . . . . . . . . . . . . . 57 3.4 The schematic configuration of a BGO anti-Compton shield of a HPGe detector. . . . . . . . . . . . . . . . . . . . . . . . . . 58 3.5 JUROGAM II array. . . . . . . . . . . . . . . . . . . . . . . . 59 3.6 Scheme of gas-filled recoil separator RITU. The figure is taken from Ref. [8]. . . . . . . . . . . . . . . . . . . . . . . . . . . . 60 3.7 Schematic diagram of GREAT sepctrometer. . . . . . . . . . . 61 3.8 Schematic illustration of a typical RDT setup and the signal times for each detector relative to the an event stamp in the DSSDs. This figure is adapted from Ref. [9]. . . . . . . . . . . 63 4.1 A sample of calibrated overlayed energy spectra of 152 Eu for some Ge detectors of the JUROGAM II array. The peaks corresponding to the contaminating transitions are indicated with asterisks. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 66 4.2 Geometry of the detector arrangement with the beam as orientation axis. . . . . . . . . . . . . . . . . . . . . . . . . . . . . Partial level scheme of 136 Nd showing the low-and medium-spin bands. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 72 5.2 Double-gated spectra for the band of 136 Nd. The peaks corresponding to the rays of 136 Nd are indicated with their energies, while those of the contaminating transitions are indicated with an asterisk. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 73 LIST OF FIGURES 5.3 Double-gated spectra for the band N2 of 136 Nd. The peaks corresponding to the contaminating transitions are indicated with asterisks. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.4 Double-gated spectrum for band L4 of 136 Nd. . . . . . . . . . 5.5 Sum of spectra obtained by double-gating on all combinations of in-band transitions of band T2. The peaks corresponding to the contaminating transitions are indicated with asterisks. . . 5.6 Partial level scheme of 136 Nd showing the dipole bands. . . . . 5.7 a) Sum of spectra obtained by double-gating on all combinations of the 220-, 254-and 294-keV transitions of band D1. b) Spectrum obtained by double-gating on the 220-and 254-keV transitions of band D1. The peaks corresponding to the in-band transitions of band D1-chiral and to the connecting transitions to band D1 are indicated with asterisks. . . . . . . . . . . . . 5.8 Spectra constructed by double-gating on transitions of band D2 which shows the connecting transitions of band D2-chiral. The transitions marked with asterisks indicate low-lying transitions in 136 Nd. The red lines show how the connecting transitions disappear when gating on successive higher-lying transitions of band D2. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.9 Double-gated spectrum on the 249-and 345-keV transitions of band D3, showing the connecting transitions of band D3-chiral to band D3, which are indicated with asterisks. . . . . . . . . 5.10 Double-gated spectrum on the 309-and 357-keV transitions of band D4. The peaks corresponding to the connecting transitions of band D4-chiral to band D4 are indicated with asterisks. . . 5.11 Sum of spectra obtained by double-gating on all combinations of the 230-, 284-, 345-, 388-and 463-keV transitions of band D5. The peaks corresponding to the in-band transitions of band D5-chiral and to the connecting transitions to band D5 are indicated with asterisks. . . . . . . . . . . . . . . . . . . . . . . 5.12 Sum of spectra obtained by double-gating on all combinations of in-band transitions of: a) band HD1 and b) band HD2. The peaks corresponding to the in-band transitions of each band are indicated with asterisks. . . . . . . . . . . . . . . . . . . . . . 5.13 Sum of spectra obtained by double-gating on all combinations of in-band transitions of: a) band HD3, b) band HD4, and c) HD5. The peaks corresponding to the in-band transitions of each band are indicated with asterisks. . . . . . . . . . . . . . 5.14 Partial level scheme of 136 Nd showing the T bands. . . . . . . 5.15 Partial level scheme of 136 Nd showing the HD bands. . . . . . 5.16 Excitation energies and h! vs I calculated by TAC-CDFT for the positive (left panel) and negative (right panel) chiral rotational bands of 136 Nd. . . . . . . . . . . . . . . . . . . . . . . . 5.17 Quasiparticle alignments calculated by TAC-CDFT for the positiveparity (left panel) and negative-parity (right panel) chiral rotational bands of 136 Nd. Solid and open circles with the same color represent experimental data of one pair of nearly degenerate bands, and different lines denote the theoretical results based on different configurations. . . . . . . . . . . . . . . . . 100 5.18 Values of transition probabilities B(M 1)/B(E2) of 136 Nd calculated by TAC-CDFT, in comparison with experimental data (solid and open symbols). . . . . . . . . . . . . . . . . . . . . 101 5.19 Evolution of the azimuth angle as a function of rotational frequency, for the total angular momentum of the configuration D ⇤ assigned to band D3, calculated by 3D TAC-CDFT . . . . 102 5.20 (Color online) The energy spectra of bands D1-D6 and their partners calculated by PRM in comparison with corresponding data. The excitation energies are relative to a rigid-rotor reference.103 5.21 (Color online) The staggering parameters of bands D1-D6 calculated by PRM in comparison with corresponding data. . . . 105 5.22 (Color online) The B(M 1)/B(E2) of bands D1-D6 and their partners calculated by PRM in comparison with corresponding data. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 106 5.23 (Color online) The root mean square components along the intermediate (i-, squares), short (s-, circles) and long (l-, triangles) axes of the rotor, valence protons, and valence neutrons angular momenta calculated as functions of spin by PRM for the doublet bands D2 and D2-C in 136 Nd. . . . . . . . . . . . 106 5.24 (Color online) Same as Fig. 5.23, but for D4 and D4-C. . . . . 107 5.25 (Color online) Same as Fig. 5.23, but for D5 and D5-C. . . . . 108 5.26 (Color online) Energies relative to a standard rotating liquid drop reference calculated for the experimental bands observed in 136 Nd. With an odd number of h 11/2 neutron holes, two signature degenerate bands are formed which are shown by the same color and symbols. . . . . . . . . . . . . . . . . . . . . . 110 5.27 The same as in Fig. 5.26 but for the bands L and T. . . . . . . 110 5.28 The same as in Fig. 5.26 but for the dipole bands. . . . . . . . 111 5.29 The same as in Fig. 5.26 but for the bands HD. . . . . . . . . 112 5.30 (Color online) The observed low-spin bands of 136 Nd are shown relative to a rotating liquid drop reference in panel (a), with the calculated configurations assigned to these bands given relative to the same reference in panel (b). The panel (c) provides the difference between calculations and experiment. . . . . . . . . 113 5.31 (Color online) The same as in Fig. 5.30 but for the medium-spin bands L. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 114 5.32 (Color online) The same as in Fig. 5.30 but for the medium-spin bands T. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 116 LIST OF FIGURES 5.33 (Color online) The same as in Fig. 5.30 but for the bands D. . 5.34 (Color online) The same as in Fig. 5.30 but for the HD bands.

  Nd. . . . . . . . . . . . . . . . . . . . . 7.2 Partial level scheme of 138 Nd related to the isomeric state. . . 7.3 Spectrum of prompt transitions measured by JUROGAM II gated with selected clean transitions in 138 Nd (521, 729, 884, and 973 keV) measured by the clovers placed at the focal plane. 7.4 Time spectra extracted from -T matrix at the focal plane transitions (521, 729, 884, and 973 keV) deexciting the 10 + isomer in 138 Nd. The red line is fitted to the data. . . . . . . . . . . .

  to the bands of 136 Nd. . . . . . . . . . . . . . . . . . . . . . . . . . .

Figure 1 . 2 :

 12 Figure 1.2: The nuclides with chiral doublet bands (red circles) and M D (blue pentagons) observed in the nuclear chart. The black squares represent stable nuclides. Figure adopted from Ref. [2].

Figure 2 . 1 :

 21 Figure 2.1: Schematic nuclear levels calculated by the shell model including the l 2 and l • s terms. Figure adapted from Ref. [3].

2. 3 .Figure 2 . 2 :

 322 Figure 2.2: The lowest four vibrations of a nucleus. The dashed lines show the spherical equilibrium shape and the solid lines show an instantaneous view of the vibrating surface. Figure adapted from Ref. [4].

Figure 2 . 4 :

 24 Figure 2.4: Schematic of nuclear shapes with respect to the deformation parameters ( 2 , ), as defined in the Lund convention. Figure adapted from Ref. [6].

Figure 2 . 8 :

 28 Figure 2.8: Schematic diagram of the strong coupling limit in the particle-rotor model.

Figure 2 . 9 :

 29 Figure 2.9: Schematic diagram of the decoupling limit in the particle-rotor model.

Figure 2 . 10 :

 210 Figure 2.10: Discrete symmetries of the mean field of a rotating triaxial reflection symmetric nucleus. The axis of rotation (z) is marked by the circular arrow. The rotational band structures associated with each symmetry type are presented on the right side. This figure was taken from Ref. [7].

Figure 3 . 1 :

 31 Figure 3.1: Schematic illustration the heavy-ion fusion-evaporation reaction forming a compound nucleus and its decay.

Figure 3 . 2 :

 32 Figure 3.2: (a) Photoelectric absorption, (b) Compton scattering, (c) Pair production.

56 3. 3 .

 563 High-purity Germanium -ray detector

Figure 3 . 3 :

 33 Figure 3.3: Reverse bias of the p-n junction.

  .4.

Figure 3 . 4 :

 34 Figure 3.4: The schematic configuration of a BGO anti-Compton shield of a HPGe detector.

Figure 3 . 5 :

 35 Figure 3.5: JUROGAM II array.

Figure 3 . 6 :

 36 Figure 3.6: Scheme of gas-filled recoil separator RITU. The figure is taken from Ref. [8].

Figure 3 . 7 :

 37 Figure 3.7: Schematic diagram of GREAT sepctrometer.

Figure 3 . 8 :

 38 Figure 3.8: Schematic illustration of a typical RDT setup and the signal times for each detector relative to the an event stamp in the DSSDs. This figure is adapted from Ref. [9].

Figure 4 . 1 :

 41 Figure 4.1: A sample of calibrated overlayed energy spectra of 152 Eu for some Ge detectors of the JUROGAM II array. The peaks corresponding to the contaminating transitions are indicated with asterisks.

Chapter 4 . 4 . 8 )Figure 4 . 2 :

 44842 Figure 4.2: Geometry of the detector arrangement with the beam as orientation axis.

Figure 5 . 1 :

 51 Figure 5.1: Partial level scheme of 136 Nd showing the low-and medium-spin bands.

Figure 5 . 2 :

 52 Figure 5.2: Double-gated spectra for the band of136 Nd. The peaks corresponding to the rays of136 Nd are indicated with their energies, while those of the contaminating transitions are indicated with an asterisk. som

Figure 5 . 3 :

 53 Figure 5.3: Double-gated spectra for the band N2 of 136 Nd. The peaks corresponding to the contaminating transitions are indicated with asterisks.

Figure 5 . 4 :Figure 5 . 5 :

 5455 Figure 5.4: Double-gated spectrum for band L4 of 136 Nd.

Figure 5 . 6 :Figure 5 . 7 :

 5657 Figure 5.6: Partial level scheme of 136 Nd showing the dipole bands.

Figure 5 . 8 :Figure 5 . 9 :

 5859 Figure 5.8: Spectra constructed by double-gating on transitions of band D2 which shows the connecting transitions of band D2-chiral. The transitions marked with asterisks indicate low-lying transitions in 136 Nd. The red lines show how the connecting transitions disappear when gating on successive higher-lying transitions of band D2.

5. 1 .Figure 5 . 10 :Figure 5 . 11 :

 1510511 Figure 5.10: Double-gated spectrum on the 309-and 357-keV transitions of band D4. The peaks corresponding to the connecting transitions of band D4chiral to band D4 are indicated with asterisks.

Figure 5 . 12 :

 512 Figure 5.12: Sum of spectra obtained by double-gating on all combinations of in-band transitions of: a) band HD1 and b) band HD2. The peaks corresponding to the in-band transitions of each band are indicated with asterisks.

5

 5

Figure 5 . 13 :Figure 5 . 14 :

 513514 Figure 5.13: Sum of spectra obtained by double-gating on all combinations of in-band transitions of: a) band HD3, b) band HD4, and c) HD5. The peaks corresponding to the in-band transitions of each band are indicated with asterisks.

Figure 5 . 15 :

 515 Figure 5.15: Partial level scheme of 136 Nd showing the HD bands.

5. 1 .

 1 Experimental results and level scheme Table

Figure 5 . 16 :

 516 Figure 5.16: Excitation energies and h! vs I calculated by TAC-CDFT for the positive (left panel) and negative (right panel) chiral rotational bands of 136 Nd.

Figure 5 . 17

 517 Figure 5.17: Quasiparticle alignments calculated by TAC-CDFT for the positive-parity (left panel) and negative-parity (right panel) chiral rotational bands of 136 Nd. Solid and open circles with the same color represent experimental data of one pair of nearly degenerate bands, and different lines denote the theoretical results based on different configurations.

  Figure 5.17: Quasiparticle alignments calculated by TAC-CDFT for the positive-parity (left panel) and negative-parity (right panel) chiral rotational bands of 136 Nd. Solid and open circles with the same color represent experimental data of one pair of nearly degenerate bands, and different lines denote the theoretical results based on different configurations.

Figure 5 . 18 :

 518 Figure 5.18: Values of transition probabilities B(M 1)/B(E2) of 136 Nd calculated by TAC-CDFT, in comparison with experimental data (solid and open symbols).

Fig. 5

 5 the orientation angle for the total angular momentum of band D3 calculated with 3D TAC-CDFT. It is found that the azimuth angle for band D3 vanishes at low rotational frequencies, providing a planar solution. Above the critical rotational frequency 0.5 MeV, a nonzero angle appears, corresponding to an aplanar solution, namely, chiral rotation. Furthermore, for band D3, the observed excitation energies and total angular momenta are also in good agreement with the 3D TAC-CDFT calculations. Although the B(M 1)/B(E2) values are not available, these results also provide very strong support for the existence of chiral rotation in band D3.Summarizing, the observed doublet bands in136 Nd have been compared with results of calculations involving adiabatic and configuration-fixed constrained CDFT, TAC-CDFT, and 3D TAC-CDFT. One can see that the observed energy spectra and alignments are well reproduced for the five pairs of nearly degenerate bands D1-D5 which exhibits the chiral character. Moreover, the band D5 and its partner B(M 1)/B(E2) values are in good agreement with the experimental data. The other four doublet bands which are weakly populated, have no measured B(M 1)/B(E2) values, but the calculated assigned configurations show chiral geometry. Hence, we suggest that D5 and D5-chiral are chiral doublet bands and the four others doublet bands are chiral candidates, which can contribute to the realization of the multiple pairs of chiral doublet bands (M D) phenomenon. It is the first time that chiral bands are observed in an even-even nucleus at high spins, and the set of five nearly degenerate bands is the largest observed in a single nucleus until now.

Figure 5 . 19 :

 519 Figure 5.19: Evolution of the azimuth angle as a function of rotational frequency, for the total angular momentum of the configuration D ⇤ assigned to band D3, calculated by 3D TAC-CDFT

Figure 5 .

 5 Figure 5.20: (Color online) The energy spectra of bands D1-D6 and their partners calculated by PRM in comparison with corresponding data. The excitation energies are relative to a rigid-rotor reference.

  Fig. 5.21 displays staggering parameters S(I) are extracted from data, compared with theoretical results. The PRM obtained values can reproduce the experimental S(I) result. Moreover, the S(I) values of all bands vary smoothly and don't change too much with spin. The behavior of S(I) represent therefore another support for the chiral interpretation of band D1-D5. The B(M 1)/B(E2) values of bands D1-D6 calculated by PRM in comparison with the corresponding experimental data are presented in Fig. 5.22.

Figure 5 .

 5 Figure 5.21: (Color online) The staggering parameters of bands D1-D6 calculated by PRM in comparison with corresponding data.

Figure 5 .

 5 Figure 5.22: (Color online) The B(M 1)/B(E2) of bands D1-D6 and their partners calculated by PRM in comparison with corresponding data.

Figure 5 .

 5 Figure 5.23: (Color online) The root mean square components along the intermediate (i-, squares), short (s-, circles) and long (l-, triangles) axes of the rotor, valence protons, and valence neutrons angular momenta calculated as functions of spin by PRM for the doublet bands D2 and D2-C in 136 Nd.

Figure 5 .

 5 Figure 5.24: (Color online) Same as Fig. 5.23, but for D4 and D4-C.

Figure 5 .

 5 Figure 5.25: (Color online) Same as Fig. 5.23, but for D5 and D5-C.

Figure 5 .

 5 Figure 5.26: (Color online) Energies relative to a standard rotating liquid drop reference calculated for the experimental bands observed in 136 Nd. With an odd number of h 11/2 neutron holes, two signature degenerate bands are formed which are shown by the same color and symbols.

Figure 5 . 27 :

 527 Figure 5.27: The same as in Fig. 5.26 but for the bands L and T.

Figure 5 . 28 : 112 5Figure 5 . 29 :

 528112529 Figure 5.28: The same as in Fig. 5.26 but for the dipole bands.

Figure 5 .

 5 Figure 5.30: (Color online) The observed low-spin bands of 136 Nd are shown relative to a rotating liquid drop reference in panel (a), with the calculated configurations assigned to these bands given relative to the same reference in panel (b). The panel (c) provides the difference between calculations and experiment.

Figure 5 .

 5 Figure 5.31: (Color online) The same as in Fig. 5.30 but for the medium-spin bands L.

Figure 5 .

 5 Figure 5.32: (Color online) The same as in Fig. 5.30 but for the medium-spin bands T.

Figure 5 .

 5 Figure 5.33: (Color online) The same as in Fig. 5.30 but for the bands D.

120 5. 2 .Figure 5 .

 12025 Figure 5.34: (Color online) The same as in Fig. 5.30 but for the HD bands.

Figure 6 . 1 :

 61 Figure 6.1: (Color online) Partial level scheme of 135 Nd showing the newly identified doublet bands.

  6.1).The spins and negative parity of band D6 are also well established based on the E2 character of the 897-, 931-, 950-, 963-, and 997-keV transitions towards band D5 (see Table.

  6.1).

Figure 6 . 2 :

 62 Figure 6.2: (Color online) Spectra for the bands D3 and D4 of 135 Nd showing the newly identified transitions (in red color).The peaks marked with an asterisk are contaminants from 136 Nd.

Figure 6 . 3 :

 63 Figure 6.3: (Color online) Comparison between the experimental excitation energies relative to a reference rotor (symbols) and the particle-rotor model calculations (lines) for the bands D1-D6.

. 5 .Figure 6 . 4 :

 564 Figure 6.4: (Color online) The experimental quasi-particle alignments for the chiral rotational bands of 135 Nd.

Figure 6 . 5 :

 65 Figure 6.5: (Color online) Comparison between experimental ratios of transitions probabilities B(M 1)/B(E2) (symbols) and the particle-rotor calculations (lines) for the bands D1-D6.

Figure 6 . 6 :

 66 Figure 6.6: (Color online) The root mean square components along the intermediate (i , squares), short (s , circles) and long (l , triangles) axes of the rotor, valence protons, and valence neutrons angular momenta calculated as functions of spin by PRM for the doublet bands D3 and D4 of 135 Nd.

Fig. 7 .

 7 1 shows a 2D -Time matrix constructed from the clovers of the focal plane in this work. One can see delayed components of different lengths for the 729-, 884-and 973-keV transitions below the 10 + , T 1/2 = 370 ns isomer of 138 Nd, and for the 640-, 815-and 948-keV transitions below the 10 + , T 1/2 = 308 ns isomer of 134 Ce.

Figure 7 . 1 :

 71 Figure 7.1: -Time matrix for the clovers at the focal plane. The transitions marked with asterisks represent the -decay contaminants from the nuclei produced in this experiment: 665 keV, 783 keV, 828 keV and 872 from the -decay of 135 Ce, 761 keV and 925 keV from the -decay of 137 Nd.

7. 3 .

 3 Results of the focal plane

Figure 7 . 2 :

 72 Figure 7.2: Partial level scheme of 138 Nd related to the isomeric state.

Figure 7 . 3 :

 73 Figure 7.3: Spectrum of prompt transitions measured by JUROGAM II gated with selected clean transitions in 138 Nd (521, 729, 884, and 973 keV) measured by the clovers placed at the focal plane.

Figure 7 . 4 :

 74 Figure 7.4: Time spectra extracted from -T matrix at the focal plane transitions (521, 729, 884, and 973 keV) deexciting the 10 + isomer in 138 Nd. The red line is fitted to the data.

  de la bande D3 et vers la bande D2-3qp est également identifié. Quatorze nouvelles transitions de 377, 389, 420, 566, 589, 612, 616, 666, 679, 705, 832, 917, 1035 et 1069 keV reliant la bande D3 à la bande D2-3qp sont également identifiées. La bande D4, connue jusqu'à l'état désexcité par la transition à 423 keV, est confirmée, mais sur la base des valeurs R DCO des transitions de connexion aux bandes D1 et D3, les spins sont diminués d'une unité et la parité est changé en positve. Nous avons ajouté un nouveau niveau avec spin 19/2 + au bas de la bande, une transition de 1237 keV vers la bande D1, quatre transitions de 596, 673, 780 et 919 keV vers la bande D3 et trois transitions de 589, 649. et 750 keV vers la bande D2-3qp. La parité de la bande D4 est identique à la parité positive assignée de la bande D3, car les trois transitions de connexion de 589, 649 et 780 keV entre les bandes 3 et 2 ont un caractère E2 bien établi. Nous confirmons tous les niveaux précédemment rapportés des bandes D5 et D6. Trois nouveaux niveaux avec les tspins 21/2 -, 45/2 -et (47/2 -) sont identifiés en bas et en haut de la bande D5, reliés par les nouvelles transitions 130, 560, 565, 1054 et 1125 keV. Les deux transitions provisoires de 557 keV et 963 keV rapportées précédemment sont confirmés, mais nos données montrent que l'énergie de la transition à 557 keV est plutôt de 559 keV. Trois nouvelles transitions reliant les bandes D6 à D5 avec des énergies de 609, 997 et 1071 keV sont récemment identifiées. Trois transitions de 463, 584 et 1299 keV des bandes D5 et D6 à la bande D1 ont également été identifiées.

Titre:

  Chiralit é dans les noyaux136 Nd et135 Nd Mots cl és : Spectroscopie nucl éaire, états de hauts spins, isom ères, triaxialit é, chiralit é R ésum é : Le mode d éxcitation collective exotique appel é chiralit é a ét é étudi é et des isom ères de longue dur ée de vie ont ét é recherch és dans les noyaux136 Nd et135 Nd.Cinq paires de bandes I = 1 presque d ég én ér ées et de m ême parit é ont ét é identifi ées à des spins élev és dans136 Nd. Les bandes observ ées ont ét é étudi ées en utilisant la th éorie de la fonctionnelle de densit é covariante contrainte et rotation autour d'une axe inclin é, et par un nouveau mod èle de rotor plus particules d évelopp é pour d écrire le couplage de particules dans quatre couches j à un rotor triaxial. Les propri ét és de ces bandes doublet sont en bon accord avec les r ésultats des calculs th éoriques. Par cons équent, l'existence de bandes chirales multiples (M D) dans le noyau136 Nd a ét é établie. Il s'agit de la premi ère preuve exp érimentale de l'existence de bandes chirales dans des noyaux pairs. Les cinq paires de bandes chirales constituent le plus grand ensamble de bandes chirales observ é jusqu' à pr ésent dans un seul noyau. De plus, le sch éma de niveaux de136 Nd a ét é consid érablement étendu aux spins bas, moyens et tr ès élev és. Les configurations possibles de toutes les bandes rotationnelles ont ét é attribu ées à l'aide du mod èle Nilsson-Strutinsky en rotation. La structure de136 Nd a ét é clarifi ée et les divers types d'excitations uni-particule et collectives ont ét é bien compris. Une nouvelle paire de bandes chirales à parit é positive a ét é identifi ée dans135 Nd. Les caract éristiques des bandes soutiennent leur interpr étation en termes de bandes chirales. Des bandes chirales à parit é négative ayant ét é identifi ées pr éc édemment, la pr ésence de multiple bandes chirales dans135 Nd est ainsi établie. Les bandes chirales observ ées ont ét é étudi ées en utilisant la th éorie avec fonctionnelle de densit é covariante contrainte et avec le mod èle rotor plus particules; elles r éproduisent fid èlement les donn ées exp érimentales, confirmant ainsi le ph énom ène M D dans ce noyau. Les bandes chirales nouvellement observ ées dans135 Nd repr ésentent une étape importante dans la confirmation de l'existence du ph énom ène M D dans les noyaux.

  1]. . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.2 The nuclides with chiral doublet bands (red circles) and M D (blue pentagons) observed in the nuclear chart. The black squares represent stable nuclides. Figure adopted from Ref. [2]. . . . .

2.1 Schematic nuclear levels calculated by the shell model including the l 2 and l • s terms. Figure adapted from Ref. [3]. . . . . . . 2.2 The lowest four vibrations of a nucleus. The dashed lines show the spherical equilibrium shape and the solid lines show an instantaneous view of the vibrating surface. Figure adapted from Ref. [4]. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.3 Plot of the Eq. 2.12, for k=1, 2, 3, corresponding to the increase in the axis lengths in the x, y, and z directions. Figure adopted from Ref. [5]. . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.4 Schematic of nuclear shapes with respect to the deformation parameters ( 2 , ), as defined in the Lund convention. Figure adapted from Ref.

  + 2n ⇢ + m l = n x + n y + n z .

	2.3. The deformed shell model
	with
	N = n z (2.23)
	)

  Nilsson diagram. Parts of Nilsson digram for neutrons and protons in the A ⇡ 130 mass region are given in Fig. 2.6 and Fig. 2.7, respectively.

										2.3. The deformed shell model
		7/2[503]							1/2[660]	3/2[651]	5/2[642]	1/2[400]	3/2[402] 3/2[402]	1/2[400]	1/2[770]	3/2[761] 1/2[640]	3/2[642] 3/2[761]	5/2[402]
	6.0 6.0 E s.p. (h -ω) s.p. (h -E ω) 1/ 2[ 42 0] 1 / 2 [ 4 1 1 ] 7 /2 [5 2 3 ] 3/2 [41 1] 3/2 [54 1] 5/2[53 2] 3/2 [42 2] 1/2 [431] 1/2 [42 0] 1 /2 [5 5 0 ] 7/2 [52 3] 1 /2 [4 1 1 ] 3 /2 [4 0 2 ] 1/2[43 1] 5/2[40 2] 9 /2 [ 5 1 4 ] 2d 3/2 2d 5/2 82 1h 11/2 2d 3/2 3s 1/2 1g 7/2 5 /2 [4 0 2 ] 3s 1/2 1h 11/2 1/ [5 50 ] 5/2[5 32] 7/2[5 23] 1 1 /2 [ 5 0 5 ] 1 / 2 [ 4 0 0 ] 3/2[ 411 ] 1 1 /2 9 /2 [ 5 1 4 ] [ 5 0 5 ] 3 /2 [4 0 2 ] 3/2[ 541] 1 1 /2 [ 5 0 5 ] 1 /2 [4 0 0 ] 1 / 2 [ 4 0 0 ] 9 /2 [5 1 4 ] 5 / 2 [ 4 0 2 ] 7 / 2 [ 4 0 4 ] 7/2 [52 3] 3/2[41 1] 5/2[41 3] 9 /2 [5 1 4 ] 1/2[411 ] 3 /2 [ 6 5 1 ] 5/2[4 13] 7 / 2 [ 4 0 4 ] 1/2[411] 3 / 2 [ 5 3 2 ] 1 / 2 [ 5 3 0 ] 1 / 2 [ 5 3 0 ] 1 /2 [6 6 0 ] 3 /2 [6 5 1 ] 3 / 2 [ 4 0 2 ] 1 /2 [ 6 6 0 ] 3 / 2 [ 5 3 2 ] 5 /2 [5 2 3 ] 9 / 2 [ 5 0 5 ] 5 / 2 [ 6 4 2 ] 1 3 /2 [ 6 0 6 ] 82 9 / 2 [ 5 0 5 ] 7 / 2 [ 5 1 4 ] 3 / 2 [ 6 5 1 ] 7 /2 [ 6 3 3 ] 1 1 / 2 [ 6 1 5 ] 3 /2 [5 2 1 ] 7 / [ 5 1 4 ] 3 /2 [5 2 1 ] 1 /2 [6 6 0 ] 1 3 /2 [ 6 0 6 ]	7 /2 [ 6 3 3 ] 1/2[411] 5 / 2 [ 6 4 2 ] 1 /2 [6 5 1 ] [ 6 4 2 ] 5 /2 1/2[411] 3/2[411] 1/2[770] 1 /2 [6 5 1 ] 5/2[523] 7/2[404] 5/2[402] 1/2[640] 7/2[633] 1/2[521]
	5.5 5.5	3 /2 [4 2 2 ]	5 / 2 [ 4 1 3 ] 7 / 2 [ 4 0 4 ] 1 1 / 2 [ 5 0 5 ] 5 /2 [ 3 ] 7 / 2 [ 4 0 4 ]	1g 7/2 2d 5/2	3 /2 [ 4 2 2 ] 1 / 2 [ 4 3 1 ]	1 / 2 [ 4 3 1 ] 1 / 2 [ 4 2 0 ] 3 /2 [ 4 2 2 ]	1 /2 [ 5 5 0 ] 3 / 2 [ 5 4 1 ] 1 /2 [ 5 5 0 ] 1 /2 [ 4 2 0 ]		3 /2 [ 5 4 ] 1	5 / 2 [ 5 3 2 ] 5 / 2 [ 5 3 2 ] 1 /2 [5 4 1 ]	3/2[651] 5/2[303] 1/2[301] 5/2[303] 1 /2 [6 6 0 ] 1 /2 [6 6 0 ] 1 /2 [5 4 1 ] 1/2[301]
	1 /2 [ 3 0 3/2[431] 1 /2 [ 3 0 1 ] 1 ] 3/2 [43 1]	1 /2 [4 4 0 ] 1/ 2[ 44 0]	50	50	7/ 2[ 41 3] 7/ 2 [4 13 ] 9 / 2 [ 4 0 4 ] 9 / 2 [ 4 0 4 ]				[ 3 0 1 ] [ 3 0 1 ] 1 /2 1 /2	3 /2	1/2[541] 5/2[532] 1/2[541] 5/2[532] 3 / 2 [ 3 0 1 ] 3/2[301]
	5.0 5.0	5/2 [4 22 ] 5/ 2 [4 22 ] 7 /2 [4 1 3 ]	7 /2 [4 1 3 ]	9 / 2 [ 4 0 4 ]	1g 9/2 1g 9/2	3 /2 [ 4 3 1 ] 5/ 2[ 42 2] 5/ 2 [4 22 ] 3 /2 [ 4 3 1 ]	5 /2 1 /2 [ 3 0 1 ] [ 3 0 3 ] 1 /2 [ 3 0 1 ]	3 /2	[ 3 0 1 ]	3 /2	[ 3 0 1 ]	3 /2 [ 5 4 1 ] [ ] 5 /2 1 [ 5 0 5 4 1 ] [5 1 /2 5 0 ]
	The Nilsson orbitals as function of the ✏ -0.2 -0.1 0.0 0.1 0.2 2 deformation are often summa-0.3 0.4 0.5 0.6 ε 2 Figure 2.6: Nilsson diagram for protons in the 50  Z  80 region showing the single-particle energies as a function of the deformation parameter ✏ 2 . For ✏ 2 >0, corresponding to the prolate shape; for ✏ 2 =0, corresponding to the spherical shape; for ✏ 2 < 0, corresponding to the prolate shape. Labels obey -0.3 -0.2 -0.1 0.0 0.1 0.2 0.3 0.4 0.5 0.6 ε 2 Figure 2.7: Nilsson diagram for neutrons in the 50  N  80 region showing the single-particle energies as a function of the deformation parameter ✏ 2 . For ✏ 2 >0, corresponding to the prolate shape; for ✏ 2 =0, corresponding to the spherical shape; for ✏ 2 < 0, corresponding to the prolate shape. Labels obey rized in a -0.3 the ⌦ ⇡ [Nn z ⇤] rule. the ⌦ ⇡ [Nn z ⇤] rule.

Table 5 .

 5

	1 -Continued

+ ! 18 +

Table 5 .

 5

	1 -Continued

Table 5 .

 5

	1 -Continued

Table 5 .

 5

	1 -Continued

Table 5 .

 5 The error on the transition energies is 0.2 keV for transitions below 1000 keV of the136 Nd reaction channel, 0.5 keV for transitions above 1000 keV and 1 keV for transitions above 1200 keV. b Relative intensities corrected for efficiency, normalized to the intensity of the 373.7 keV transition. The transition intensities were obtained from a combination of total projection and gated spectra. c R DCO has been deduced from an asymmetric -coincidence matrices sorted with detectors at 157.6 on one axis, and detectors at ⇡ 90 on the other axis. The tentative spin-parity of the states are given in parenthesis. d R ac has been deduced from two asymmetric -coincidence matrices sorted with all detectors at 133.6 and 157.6 on one axis, and detectors at ⇡ 90 on the other axis. The tentative spin-parity of the states are given in parenthesis.

	1 -Continued

a e DCO ratio from spectrum gated on stretched quadrupole transition. f DCO ratio from spectrum gated on stretched dipole transition.

Table 5 .

 5 2 -Experimental information including the ray energies, energies of the initial levels E i and the tentative spin-parity assignments to the observed states in

Table 5 .

 5 3 -Unpaired nucleon configurations labeled A-H and the corresponding parities, calculated by constrained CDFT. The excitation energies E x (unit MeV) and quadrupole deformation parameters ( , ) are also presented.

	State E x Parity	( , )	U n p a i r e d n u c l e o n s
	G 0.000 +	(0.24, 27 )	-
	A 0.335 + B 3.419 + C 3.704 + D 1.173	(0.21, 21 ) ⇡(h (0.22, 19 ) ⇡(h (0.26, 23 ) ⇡(h 11/2 ) 1 (d 11/2 ) 3 (d 11/2 ) 2 (g 5/2 g 5/2 g 7/2 ) 2 ⌦ ⌫(h 7/2 ) 1 ⌦ ⌫(h 7/2 ) 1 ⌦ ⌫(h 11/2 ) 1 (f 11/2 ) 1 (s 11/2 ) 1 (s 7/2 h 1/2 d 1/2 d 9/2 ) 1 3/2 ) 1 3/2 ) 1 (0.22, 19 ) ⇡(h 11/2

  /2 ) n 4 , for which we will use the short hand notation [(p

					1 )p	2 p	3 , n	1 n	2 (n	3 n	4 )]. The
	pseudo-spin partners d	5/2 g	7/2 (dg), s	1/2 d	3/2 (sd) and h	9/2 f	7/2 (hf )

Table 5 .

 5 [START_REF] Greiner | Nuclear models[END_REF] -Configuration assignments and deformation information to the bands of136 Nd.

	Band I(%) Parity	Configuration	States	("	2 , )
	GSB 100 -band 6 N1 9 N2 5 L1 34 L2 6 L3 2 L4 3 L5 1.4 L6 1.2 L7 7.5 L8 0.2 L9 0.7 T1 3	+ + --+ + + + --+ + + +	[10 0,42] [82,42] [9 +, 1 ,42] [10 0, 3 3 +, ] [82,42] [82,42] [9 1 ,3 + 3 ] [82,42] [7 + 3 + ,42] [7 3 + ,42] [10 0,24] [9 1 + ,3 3 ] [82,24] [7 3 ,3 3 ]	0 + -10 + 2 + -20 + 5 -20 6 -14 10 + -26 + (⇡0.20, ⇡25 ) (⇡0.16, ⇡-20 ) (⇡0.20, ⇡26 ) (⇡0.18, ⇡25 ) (⇡0.16, ⇡-25 ) 14 + -20 + (⇡20, ⇡25 ) 13 + -21 + (⇡0.16, ⇡-25 ) 12 + -22 + (⇡0.20, ⇡25 ) 17 -23 (⇡0.20, ⇡25 ) 16 -20 (⇡0.20, ⇡20 ) 10 + -16 + (⇡0.15, ⇡-35 ) 13 + -18 + (⇡0.17, ⇡-33 ) 18 + -22 + (⇡0.18, ⇡27 ) 20 + -32 + (⇡0.18,⇡ -35 )
	T2 T3 T4 D1 D2 D3 D4 D5 D6 HD1 HD2 0.6 0.7 0.6 0.7 1.7 1.9 2.6 2.9 8.4 1.2 2 HD3 0.4 HD4 0.1 HD5 0.3	---+ + --+ -[7 [82,3 3 [82,3 [82,3 3 ] + 3 + ] [7 + 3 ,24] [9 1 +, ,3 + 3 ] [7 + 3 + ,3 3 +, ] [82,3 + 3 +, ] [82,3 3 +, ] +, (1 0)] + 3 + ,3 3 +, (1 0)] 21 -31 15 -(39 ) (⇡0.19, ⇡26 ) 26 -(37 ) (⇡0.17, ⇡ -30 ) 24 -(38 ) (⇡0.20, ⇡-70 ) 11 + -20 + (⇡0.18, ⇡-25 ) 15 + -26 + (⇡0.20, ⇡21 ) 13 -23 (⇡0.20, ⇡25 ) 14 -25 (⇡0.20, ⇡25 ) 15 + -29 + (⇡0.23, ⇡27 ) (⇡0.23, ⇡25 ) + [64,5 4(21)] (17 + )-(53 + ) (⇡0.29, ⇡5 ) + [64,5 + 4(21)] (18 + )-(52 + ) (⇡0.29, ⇡5 ) -[64,45 (21)] (25 )-(47 ) (⇡0.27, ⇡15 ) -[64,45 + (21)] (28 )-(50 ) (⇡0.27, ⇡15 ) -[64,44(1 + 1)] (19 )-(49 ) (⇡0.27, ⇡17 )

  + , which together with the previously known 39/2 + level form a new band labelled D2-5qp. The spins of band D2 are established based on the dipole character of the several connecting transitions to band D1. The parity of band D2 is not experimentally Chapter 6. Evidence of M D in the odd-A nucleus135 Nd

							125
	D5 (47/2 )	565 45/2 1125	1054 43/2 560 D6	291 418 248 668 539 291 173 649 358 764 406 845 226 282 309 438 372 901 591 963 251 565 170 121 25/2 29/2 33/2 37/2 462 440 935 21/2 714 584 130 559 444 1299 23/2 27/2 31/2 35/2 39/2 472 512 966 41/2 494 609 1071 997 950 640 931 649 897 671 841 463 592 27/2 31/2 39/2 35/2 812 953 29/2 33/2 37/2 (41/2 ) 681
	836 585 855 204 270 192 208 243 889 1161 251 177 396 451 381 400 830 544 301 670 369 759 390 1074 192 1015 236 285 334 328 832 589 705 389 566 1184 236 268 344 423 876 486 911 419 649 197 1237 750 944 835 735 19/2 23/2 27/2 31/2 425 1032 938 668 748 276 D1 541 962 37/2 966 (1024) 17/2 21/2 21/2 23/2 25/2 29/2 19/2 25/2 29/2 D4 27/2 596 589 673 780 919 520 663 21/2 29/2 (33/2 ) D3 403 19/2 23/2 25/2 27/2 31/2 33/2 580 377 420 612 917 616 666 1035 679 1069 17/2 21/2 25/2 29/2 33/2 422 839 39/2 D2-3qp 417 814 41/2 398 834 43/2 436 Nd 135 (45/2 ) D2-5qp	560 594 362 233 11/2 9/2 199 13/2 709 477 727 1217 15/2
				35/2	31/2	27/2	23/2	19/2	15/2

  133 Ce,135 Nd,

	nuclei.
		In summary, a new pair of chiral doublet bands based on the 3qp ⇡[(1h	11/2 ) 1
	(1g	7/2 ) 1 ]⌦⌫(1h	11/2 ) 1 in addition to the previously known pair of chiral bands
	based on the ⇡(1h	11/2 ) 2 ⌦⌫(1h	11/2 ) 1 configuration has been identified in

Table 6 . 1 -

 61 ContinuedE a

  DCO has been deduced from asymmetric -coincidence matrix sorted with detectors at 157.6 on one axis, and detectors at ⇡ 90 on the other axis. The tentative spin -parity of the states are given in parenthesis. d R ac has been deduced from two asymmetric -coincidence matrices sorted with detectors at 133.6 and 157.6 on one axis, and detectors at ⇡ 90 on the other axis, respectively. The tentative spin -parity of the states are given in parenthesis.

e DCO ratio from spectrum gated on stretched quadrupole transition. f DCO ratio from spectrum gated on stretched dipole transition.

e 2Rc .

3.7. Total-Data Readout (TDR)

a The error on the transition energies is 0.2 keV for transitions below 1000 keV of the[START_REF] Simon | Evidence for nuclear shape coexistence in 180 Hg[END_REF] Nd reaction channel, 0.5 keV for transitions above 1000 keV and 1 keV for transitions above 1200 keV. b Relative intensities corrected for efficiency, normalized to the intensity of the 198.8 keV transition. The transition intensities were obtained from a combination of total projection and gated spectra.

7.3. Results of the focal plane

Acknowledgements

I would like to express my sincere gratitude to those who helped me and shared their experiences during my PhD. The work presented here would have been impossible to conclude without their guidance and assistance.

I would like to also thank the financial support of China Scholarship Council (CSC) and CSNSM, CNRS.

To bands T3 and T4, the possible configuration [82, 3 3 ] and [7 + 3 , 24] have been assigned, respectively. Their quadrupole deformations decrease gradually with increasing spin at quasiconstant triaxiality, from (" 2 ⇡ 0.20, ⇡ 70 ) to (" 2 ⇡ 0.12, ⇡ 68 ) for band T3, and from (" 2 ⇡ 0.17, ⇡ 30 ) to (" 2 ⇡ 0.12, ⇡ 45 ) for band T4.

The dipole bands

The CNS configurations assigned to the dipole bands of 136 Nd are presented in Table. [START_REF] Greiner | Nuclear models[END_REF].5, which are in global agreement with those calculated with the CDFT model in Ref. [START_REF] Petrache | Evidence of chiral bands in even-even nuclei[END_REF] and with the PRM in Ref. [55]. In the CNS model, we can only describe the configurations for yrast partners of the chiral doublets, which assumes the rotation around one of the principal axes. As one can see in Table 5.5, all dipole bands D1-D6 have a pronounced triaxiality, close to the maximum of 30 . The calculated triaxiality is positive for all bands excepting for band D1. The configuration assignment is quite straightforward, being based on the measured energies, spin-parities and decay patterns. The configuration assignments are as follows:

The positive-parity band D1 decays to band L7 and the bottom of band L1, which are based on ⌫h 2 11/2 and ⇡h 2 11/2 configurations, respectively. The assigned [9 1

+, , 3

+ 3 ] configuration is the simplest single-particle excitation leading to a low-lying positive-parity band, which, however maintain one low-⌦ proton and one high-⌦ neutron in the h 11/2 orbitals to assure the perpendicular geometry of the angular momenta required by dipole and chiral bands.

The positive-parity band D2 is developed above spin I = 15 + and has a fragmented decay to many low-lying bands, including band D1. A twoquasiparticle excitation with respect to band D1 appears as the natural choice, and we therefore assign the [7 + 3 + , 3 3 ) intruder orbital relative to band L6. As one can see in the panel (c) of Fig. 5.33, the data are 6.3. Discussion and to the variation of the moment of inertia which is not taken into account in the calculation. The corresponding calculated B(M 1)/B(E2) ratios, shown in Fig. 6.5 (a), are in very good agreement with the measured values.

The configurations assigned to band D2-3qp and its continuation at high spins D2-5qp are ⇡[(1h

) 1 , respectively. The deformation parameters obtained from the CDFT calculations are ( 2 , ) = (0.23, 22.5 ) and (0.25, 15.6 ), respectively. To reproduce the B(M1)/B(E2) for D2-3qp, a slightly smaller triaxial deformation 17.0 is used in the PRM calculations. In both calculations, a Coriolis attenuation factor ⇠ = 0.92 is introduced. In addition, the used moments of inertia for D2-3qp and D2-5qp are J o = 25.0 and 40 h2 MeV 1 , respectively. A larger J o needed for D2-5qp is consistent with its lager single-particle alignment, as shown in. Fig. 6.2. As one can see in Fig. 6.2, the calculated excitation energies for band D2 are in excellent agreement with the experimental values in the central and high spins ranges. At low spins, the calculated energies are lower than the experimental values by around 200 keV, which can be ascribed to the interaction with other low-lying levels. The calculated B(M 1)/B(E2) ratios of band D2 are compared with the experimental data in Fig. 6.5 (b), in which we can see a good agreement for both D2-3qp and D2-5qp bands.

The configuration assigned to the positive-parity doublet bands D3 and D4 is ⇡[(1h

, similar to that assigned to the corresponding bands 2 and 3 of the isotone nucleus 133 Ce [START_REF] Ayangeakaa | Evidence for multiple chiral doublet bands in 133 Ce[END_REF]. In the PRM calculations, ( 2 = 0.23, = 21.0 ), J 0 = 28 h2 MeV 1 , and ⇠ = 0.94 are employed. The calculated energy spectra presented in Fig. 6.3 reproduce well the experimental data. The energy separation between the bands is nearly constant at ⇡ 200 keV, reflecting similar moments of inertia, and supporting thus the chiral doublet bands interpretation. Due to very weak in-band crossover transitions, we could extract the B(M 1)/B(E2) ratio only for the level with spin I = 29h of band D3, which is similar to that of band D2. The theoretical calculation is in agreement with the experimental values within the error bars [see Fig. 6.5 (b)]. Based on this similarity between the observed bands in the two isotones 133 Ce and 135 Nd, we safely can interprete the bands D3 and D4 of 135 Nd as chiral doublet bands based on the ⇡[(1h

The configuration adopted for the bands D5 and D6, which are among the best examples of chiral vibration, is ⇡(1h 11/2 ) 2 ⌦ ⌫(1h 11/2 ) 1 [START_REF] Mukhopadhyay | From chiral vibration to static chirality in 135 Nd[END_REF][START_REF] Zhu | A composite chiral pair of rotational bands in the odd-A nucleus 135 Nd[END_REF]51]. The deformation parameters from the CDFT are ( 2 , ) = (0.24, 22.2 ), which which are the same as those used in Ref. [START_REF] Mukhopadhyay | From chiral vibration to static chirality in 135 Nd[END_REF]. To reproduce the rapid increase of experimental B(M 1)/B(E2) value at I = 39/2 h, a slightly smaller triaxial deformation = 20.2 is used in the PRM calculations. In addition, J 0 = 23.5 h2 MeV 1 , and ⇠ = 0.98 are adopted. As one can see in Fig. 6.3, the calculated excitation energies of the two bands are in very good agreement with the experimental data. The measured transition probabilities B(M 1) and B(E2) of the two bands were published in Ref. [START_REF] Mukhopadhyay | From chiral vibration to static chirality in 135 Nd[END_REF]. The B(M 1)/B(E2) ratios of the Appendix A The ✓ is defined with respect to the beam direction. c = 0 is defined as vertically upwards, increases in a clockwise direction when the array is viewed from a position upstream.

JUROGAM II detector angles