Tables of contents Introduction

The nuclear shape coexistence is a spectacular and universal phenomena reflecting the competition between different microscopical configurations at low excitation energy corresponding to different nuclear shapes in a single nucleus [START_REF] Heyde | Shape coexistence in atomic nuclei[END_REF]. In the nuclear chart, in most of the case, the nuclear shape coexistence involves a spherical configuration that coexistences with a deformed configuration. The corresponding states reflect the competition between spherical gap from major or sub-shell closure and the residual energy of the nuclear interaction which minimize the energy of the deformed configuration. The ground state and the low lying excited states corresponding to different shapes coexist in the same nucleus with a difference in excitation energy typically lower than 2 MeV. It illustrates the subtile equilibrium of the nuclear interaction near the ground state. This phenomena is therefore a strong constraint for modern nuclear models and its experimental description remains a challenge involving advanced heavy-ions facilities and experimental setup. In a unified description of shape coexistence [START_REF] Heyde | Shape coexistence in atomic nuclei[END_REF] for nuclei close to proton shell closures, e.g. Z=20 (Ca [START_REF] Caurier | The shell closure; from to the neutron drip line[END_REF]), 28 (Ni [START_REF] Pauwels | Pairing-excitation versus intruder states in 68 Ni and 90 Zr[END_REF]), and 82 (Pb-Hg [START_REF] Nazarewicz | Variety of shapes in the mercury and lead isotopes[END_REF]), spherical configurations corresponding to closed shells compete with deformed configurations resulting from multi-particle multi-hole excitations above the proton shell closures that interact with neutrons which fill high-j intruder orbits. The significant gain in binding energy for this configuration arising from the pairing and the proton-neutron correlations energy decreases the excitation energy of the corresponding 0 + 2 state in even-even nuclei so that it appears at low excitation energy or even becomes the ground state [START_REF] Heyde | Shape coexistence in atomic nuclei[END_REF]. Shape coexistence has been the subject of numerous theoretical works and almost all theoretical approaches have been compared with available experimental data. Recently, a major achievement was reached by the use of Monte Carlo Shell Model calculations. These calculations provide for the first time a microscopic description in the neutron rich Ni and Zr isotopes and in neutron deficient Hg isotopes with an accurate reproduction of the experimental data [START_REF] Togashi | Quantum phase transition in the shape of zr isotopes[END_REF][START_REF] Tsunoda | Novel shape evolution in exotic ni isotopes and configuration-dependent shell structure[END_REF][START_REF] Marsh | Characterization of the shape-staggering effect in mercury nuclei[END_REF][START_REF] Singh | Evidence for coexisting shapes through lifetime measurements in 98 Zr[END_REF]. A second achievement is the first measurements of spectroscopic quadruopole moment of short lived excited states using post-accelerated radioactive beams at ISOL facilities and safe Coulomb excitation technique [START_REF] Clément | Shape coexistence in neutron-deficient krypton isotopes[END_REF][START_REF] Clément | Spectroscopic quadrupole moments in 96,98 Sr: Evidence for shape coexistence in neutron-rich strontium isotopes at n = 60[END_REF][START_REF] Clément | Low-energy coulomb excitation of 96,98 Sr beams[END_REF][START_REF] Bree | Shape coexistence in the neutron-deficient even-even 182-188 hg isotopes studied via coulomb excitation[END_REF]. Very recently, single nucleon transfer shed a new light on the field [START_REF] Cruz | Shape coexistence and mixing of low-lying 0+ states in 96sr[END_REF]. The last decade has also shown that this exotic phenomena is not rare. From isolated case, islands of occurrence appeared all along the nuclear chart, at proton shell closures closed by high j-orbits, involving therefore a possible strong proton-neutron interaction via the tensor force. Finally, some cases appeared in the recent years close to doubly magic nuclei ( 42 Ca [START_REF] Hadyńska-Klȩk | Quadrupole collectivity in 42 Ca[END_REF], 80 Ge [START_REF] Gottardo | First evidence of shape coexistence in the 78 Ni region: Intruder 0 + 2 state in 80 Ge[END_REF], 56 Ni [START_REF] Rudolph | Rotational bands in the doubly magic nucleus 56 ni[END_REF], 78 Ni [START_REF] Taniuchi | 78ni revealed as a doubly magic stronghold against nuclear deformation[END_REF]).

The experimental study of shape coexistence is long standing effort. Many experimental probes can be used to discover or characterized shape coexistence in an isotopic chain or in a given nuclei. Shape changes such as the transition from spherical to deformed nuclei can be identified by the measurement of the excitation energy of the first excited 2 + state in even-even nuclei. In odd-mass nuclei, having a non zero spin in the ground state, the charge radii or the spectroscopic quadrupole moment extracted from laser spectroscopy, are a clear signature of a shape change. The most emblematic case is the discovery of the charge radii staggering in the Hg isotopes [START_REF] Kühl | Nuclear shape staggering in very neutron-deficient hg isotopes detected by laser spectroscopy[END_REF][START_REF] Bonn | Sudden change in the nuclear charge distribution of very light mercury isotopes[END_REF][START_REF] Marsh | Characterization of the shape-staggering effect in mercury nuclei[END_REF] between the ground state and the isomeric state, suggesting a shape coexistence between spherical and deformed configuration. In heavy elements, α-decay have been used to identify excited 0 + shape isomer in the light Pb isotopes [START_REF] Andreyev | A triplet of differently shaped spin-zero states in the atomic nucleus 186pb[END_REF]. E0 0 +m → 0 + transition by e -decay is a unique signature of shape coexistence [START_REF] Van Duppen | Observation of low-lying J π = 0 + states in the single-closed-shell nuclei 192-198 Pb[END_REF][START_REF] Bouchez | New shape isomer in the self-conjugate nucleus 72 Kr[END_REF][START_REF] Kawade | Excited 0+-levels in98zr[END_REF][START_REF] Lhersonneau | Evolution of deformation in the neutron-rich zr region from excited intruder state to the ground state[END_REF][START_REF] Schussler | Discovery of a very low-lying 0+ state in 98sr and shape coexistence implication in 98sr[END_REF]. A large value of the monopole transition strenght ρ 2 (E 0 ) reflects a large difference in deformation or/and a large mixing of the 0 + states wave function [START_REF] Heyde | Monopole strength as a measure of nuclear shape mixing[END_REF]. This large mixing can lead to the repulsion of the 0 + states and increase the excitation energy of the 0 + 2 state. If both the monopole and quadrupole strenghts are known [START_REF] Bengtsson | Some results from potential energy surface calculations for nuclei in the mass 70-80 region[END_REF], the mixing amplitude of the wave functions can be deduced. This strenght distribution related to a significant mixing perturbates the relative excitation of the first 2 + to the ground state (that increases) and the reduced E2 transition probability (which is reduced) from an extrapolation of the collective band at higher angular momentum. This low spin perturbation was interpreted as a signature of mixing related to shape coexistence. A complete illustration is the case of the neutron deficient Kr isotopes [START_REF] Kelsall | [END_REF]29,30,31,[START_REF] Görgen | Lifetime measurement in 74kr and 76kr[END_REF]. Ideally, one would like to measure directly the shape of both configurations. The quantities directly related to the charge distribution are the charge radii and the electric spectroscopic quadrupole moment. In odd mass nuclei, with states that differ from zero spin, such measurement can be achieved by laser spectroscopy under the condition that both the ground state and the shape isomer live long enough for the measurement. Deformation parameter for a 0 + state cannot be directly measured since the zero spin prevents to define an orientation needed to define a shape. First 2 + excited states built on top of the corresponding 0 + state live from few pico-second to few nano-second. Laser spectroscopy remains impossible. The last decade has revolutionized the study of shape coexistence with the possibility of having post-accelerated radioactive beam at the Coulomb barrier in so-called ISOL facilities like GANIL-SPIRAL1, REX-ISOLDE and ISAC-TRIUMF. With beams intensity > 10 3 pps and efficient γ arrays like EXOGAM [START_REF] Simpson | The exogam array: A radioactive beam gamma-ray spectrometer[END_REF], MINIBALL [START_REF] Warr | The miniball spectrometer[END_REF] and TIGRESS [START_REF] Svensson | Position sensitivity of the tigress 32fold segmented hpge clover detector[END_REF], multistep safe Coulomb excitation [START_REF] Alder | The theory of coulomb excitation of nuclei[END_REF][START_REF] Alder | Theory of coulomb excitation[END_REF][START_REF] Alder | Theory of Coulomb excitation with Heavy ions[END_REF][START_REF] Cline | Nuclear shapes studied by coulomb excitation[END_REF]40,[START_REF] Zielińska | Analysis methods of safe coulombexcitation experiments with radioactive ion beams using the gosia code[END_REF] becomes feasable. From these measurements, using second order effects in the Coulomb excitation cross section, the diagonal E2 matrix element J π |E2|J π can be measured in short lived excited states. This quantity is directly related to the spectroscopic quadrupole moment (Q s ) and its sign. A geometrical conversion allows to determine an axial deformation parameter β 2 . Thus such measurements allow to determine the corresponding shape of excited states and therefore the unique way to prove shape coexistence in exotic nuclei involving short lived non-zero spin states.

Shape coexistence with protons and neutrons far from shell closures have been reasonnably well describes within the mean-field approaches using interactions like the Gogny or Skryme functionnals. In these calculations, two clear minima appear in the potential energy surface calculated as a function of the deformed axial (β) and triaxial (γ) degree of freedom. Each minima corresponds to a J π = 0 + state in even-even nuclei, from which collective structures are built. In the vicinity of proton major or sub-shell closure, a more microspical description of the low lying 0 + state was developped by K. Heyde and co-workers [START_REF] Heyde | A shell-model description of 0+ intruder states in even-even nuclei[END_REF]. Starting from the nuclear shell structure in medium-heavy and heavy nuclei, the excitation energy for low-lying 0 + intruder states was investigated. Taking a simplified model with two particle-two hole (2p-2h) excitations across closed shells, the effects of pairing and proton-neutron (monopole and quadrupole components) with high-j in-truder orbits residual interaction was highlighted. The resulting contribution of the pairing and quadrupole proton-neutron interaction allow the deformed state to gain enough binding energy to become the ground state. The spherical configuration doesn't vanish but appears as an excited configuration. It is a caracteristic of the shape coexistence phenomenum that it is not only an onset of deformation (as for N=90) or the local appearence of a spherical shape (as magic nuclei). But both configurations, spherical and deformed, coexist at low excitation energy and are in competition for the ground state configuration. Only subtle effects in the nuclear interaction make one or the other configuration a candidate for the ground state as minimum of the potential energy of the system. The microscopical description calls for more elaborated calculations. This was recently achieved using Monte Carlo Shell Model calculation in several region benchmarking the phenomenom of shape coexistence [START_REF] Togashi | Quantum phase transition in the shape of zr isotopes[END_REF][START_REF] Tsunoda | Novel shape evolution in exotic ni isotopes and configuration-dependent shell structure[END_REF][START_REF] Marsh | Characterization of the shape-staggering effect in mercury nuclei[END_REF][START_REF] Singh | Evidence for coexisting shapes through lifetime measurements in 98 Zr[END_REF]. However, it is worth mentioning that, in spite of all these great progresses, the comparaisons with spectroscopic data are always partial. Indeed, most of the advanced theoretical models are compared with experimental data of a given publication. Often, partial level schemes are compared on few isotopes and time to time reduced transition probabilities. Rarely, spectroscopic quadrupole moment from both excited states and ground states are compared at the same level. Even more rarely, monopole transition strenght ρ 2 (E 0 ), often evidencing for the first time shape coexistence by the observation of a 0 + → 0 + transition, are systematically compared. It can also be underligned that so far, the so-called ab -initio calculations did not yet investigated such shape competition. It will be interesting to evaluate if such spectacular phenomena emerge from f irst-principle approaches giving more insight in the microscopical origin of the deformation in atomic nuclei and by extension to shape coexistence. Finally, symmetries can play a major role. The islands of inversion have also been described in the Quasi-SU [START_REF] Pauwels | Pairing-excitation versus intruder states in 68 Ni and 90 Zr[END_REF] and Pseudo-SU(3) dynamical symmetries framework [43,44] and can be applied to many area of shape coexistence.

The sub-shell closure at nucleons number 40 plays a major role in the framework of the study of the shape coexistence. This sub-shell closure is not a strong shell closure defining magic or semimagic nuclei as can be clearly seen from the mass measurements. Its appears as the 4 th major shell gap in the oscillator potential plus angular momentum but without spin-orbit. It is closing the spdf sequence, last orbit beeing the l=2 2p, by the l=4 1g orbit. This natural ∆=2 difference calls for large quadrupole collectivity effect when nucleons are excited from the spdf shells to the g orbit. Thanks to the spin-orbit term, the 1g orbit is further splitted between the 1g 9/2 (down in energy) and the 1g 7/2 (increased energy) which define finally the strong N, Z = 50 shell closure responsible for the doubly magicity character of 100,132 Sn and 78 Ni isotopes. This proximity in energy between the 1g 9/2 orbit and the pf shells is the seed for the competition between spherical states with nucleons occupying the pf shells and deformed configurations involving nucleons excitation into the 1g 9/2 orbit. The occupancy of the g-orbit only doesn't provide sufficient gain in binding energy to allow the deformed configuration to become the ground state. Similarly to the N = 20 and N = 28 island of inversions, thus shape coexistence, are triggered by a proton-neutron interaction reducing the spin-orbit splitting induced by the tensor term of the interaction in its ∆l=0 or ∆l=1 channels. Proton-neutron interactions are the final trigger of shape coexistence provinding enough correlation energy to reduce the excitation energy of the deformed configuration. At N = 20, the p -n ∆l=0 interactions are πd 5/2 -νd 3/2 and πd 3/2 -νd 3/2 as at N = 28, the ∆=1 interactions are πd 5/2 -νf 7/2 and πd 3/2 -νf 7/2 . This scheme can also be applied to isotopes involving either a proton or a neutron number 40 or both a proton and a neutron number 40 making this nucleon number unique in the nuclear chart.

At Z∼ 40 and N∼60, the π(g 7/2 -g 9/2 ) spin-orbit splitting is reduced when neutrons are promoted into the νg 7/2 and less significantly into the νh 11/2 orbits from the νd 5/2 orbit. Neutron holes interact strongly in the ∆l=1 channel with proton holes in the πf 5/2 orbit created by pair excitation across the Z=40 gap into the πg 9/2 orbit. This proton pair further interacts with a pair of promoted neutron in the νg 7/2 orbit in the ∆l=0 channel. Theses strong p-n interactions induce the sudden and unique onset of deformation leading to shape coexistence at N=60 in neutron rich Sr/Zr elements which desappears for elements Z > 40 and Z < 37. My contribution in this mass region is described in chapter II.

At Z∼28 and N∼ 40, the π(f 5/2 -f 7/2 ) spin-orbit splitting is reduced when neutron are excited across the N=40 gap and populate the νg 9/2 orbit from the νf 5/2 orbit. Neutron holes interact strongly in the ∆l=0 channel with proton holes in the πf 7/2 orbit created by the pair excitation across the Z=28 gap into the πp 3/2 orbit. The resulting deformed configuration competes at ∼ 2MeV excitation energy with the spherical ground state in the 68 Ni isotope, and is almost degenerated in the 67 Co isotopes (Z=27) to become the ground state in the Z=26 66 Fe nuclei. My contribution in this shape coexistence scenario is described in chapter III.

At Z∼N∼40, the description in term of particle-hole excitation of proton interacting with neutron populating high-j intruder orbits is not directly applicable. Being at and near the N=Z line, neutron-deficient isotopes of Ge, Se, Kr, Sr and Zr do not follow the large isospin asymmetry leading to shape coexistence with respect to the two previous cases. Both proton and neutron occupy same orbits and have similar occupancy of the g 9/2 orbit. This isospin symmetry seems to inihibit the competition between spherical and deformed configuration. A shape coexistence scenario between spherical and deformed states was proposed in 70,72,74 Ge and will not be discussed here. In neutron deficient, Se, Kr, Sr and Zr, prolate and oblate configurations compete for the ground state giving rise to a unique prolate-oblate shape coexistence scenario with very large mixing of the wave functions by opposition to the previous cases. The nearly identical microscopic configuration of both proton and neutron arise from the fact that naturally proton and neutron occupy same orbits. By opposition to the A∼100 and 68 Ni regions where a proton-neutron interaction drives the shape competition, the N∼Z region could be explained by mean field effects. The weak occupancy of the g 9/2 orbit gradually increases along the N=Z line and can be correlated to the level of triaxility of the ground state. My contribution in this shape coexistence scenario is described in chapter IV.

Partie I

Avant-Propos

Chapter 1

Avant-Propos

One of the key parameter in the description of the nucleus is its mass distribution. Since the first observations of rotational bands similar to molecules in nuclei, it is well established that they are breaking the spherical symmetry and get deformed. Almost all nuclei are far from the spherical shape. This deviation is called nuclear deformation. The most simplest nucleus, 2 H, has a non-zero quadrupole moment which indicates that on average a preference for a non-spherical shape for the nuclear interaction. The question of the description of the nuclear deformation, both theoretically and experimentally, remains a challenge in nuclear physics. On the theoretical side, one could summarize the open question to which degrees of freedom are relevant for the description of the nuclear shape. This lead to the development of non-spherical mean-field model at different orders or group-symmetry based approaches. The dominant symmetry is the axial symmetry with the so-called prolate and oblate deformation. Smaller corrections on the potential energy can be obtained by including the triaxial or the octupole degree of freedom.

Proton and neutrons create electric and magnetic fields in their orbits inside the nucleus. The V potential created by the charge distribution can be written in multi-pole moments such as :

V (R) = 1 R ρ(r)dr Electric monopole + 1 R 2 zρ(r)dr Dipolar term + 1 R 3 (3z 2 -r 2 )ρ(r)dr Quadrupolar term +... (1.1)
The nuclear surface R can be written using the spherical shape corrected by the normalized spherical harmonic such as :

R(θ, φ) = R 0 (1 + λ=0 λ µ=-λ α λµ Y λµ (θ, φ)) (1.2)
The λ, µ indexes defined the degree of symmetry of the nuclear surface. When limited to the triaxial degree of freedom, the three non null terms are α 2,0 , α 2,2 and α 2,-2 . The later can be written as a function of two deformations parameters β and γ defined as :

α 20 = βcosγ α 22 = α 2-2 = 1 √ 2 βsinγ (1.3)
β represents the axial elongation and γ is the asymmetry also called the triaxial degree of freedom. Using the ellipsoidal description of the nuclear volume, β is proportional to the difference between the long and short axis such as

β = 4 3 π 5 c-a R .
The comparison of measured and calculated nuclear deformation and its evolution with mass, charge and angular momentum is a cornerstone of the nuclear researches. On the experimental side, there are many methods to estimate the nuclear deformation. However measuring directly β or γ is not possible. From the observation of excited states, one can deduce from geometrical models an axial deformation parameter β using for instance the excitation energy of the first 2 + state. In collective models, the reduced transition probabilities measures the wave functions overlap between different excited states under a given electromagnetic operator. For instance the B(E2, 2 + 1 →0 + 1 ) gives the wave function overlap between the 2 + 1 and 0 + 1 states using the electric quadrupole operator. For all multi-polarities, the reduced transition probability is proportional to the matrix element between states and can be written such as :

B(Eλ, I i → I f ) = | I f || M (Eλ) || I i | 2 2I i + 1 (1.4)
In deformed nuclei, the B(E2)'s are large, underlining that excited states have the same structure but differ by their angular momentum. Similarly, under the assumption of geometrical models, a β parameter can be deduced from the B(E2, 2 + 1 →0 + 1 ).

The most accurate quantity to determine the nuclear deformation is the electric spectroscopic quadrupole moment. Its amplitude determines the overall deformation and its sign the prolate or oblate character. Similarly to the reduced transition probability, the spectroscopic quadrupole moment is proportional to the mean value of the E2 operator for a given state such as, in the laboratory frame :

eQ s = 16π 5 (2I + 1) -1/2 II20 | II I || M (E2) || I (1.5)
This quantity can be translated in the nucleus reference frame (intrinsic quadrupole moment) by :

eQ 0 = 16π 5 1 √ 2I i + 1 I f M (E2) I i I i K20|I f 0 (1.6)
With I f = I i , one deduces a transitional quadrupole moment which is related to the reduced transition probability by :

B(E2, I i → I f ) = 5 16π (eQ 0 ) 2 I i K20|I f 0 2 (1.7)
With I f = I i , the static quadrupole moment (Q 0 ) can be deduced and can be translated in β such as :

Q 0 = 3 √ 5π ZR 2 β 1 + 1 8 5 π β (1.8)
A direct translation from the intrinsic to the laboratory frame is given by :

Q 0 = 3K 2 -I(I + 1) (I + 1)(2I + 3) Q s
In excited states, the reduced transitional probability can be deduced from the measurement of their lifetime such as :

1 τ = 12.26B(E2, ↓)E 5 γ (1 + α) τ is in [ps], the B(E2) in [e 2 f m 4 ] and E γ in [MeV]
. The α coefficient is the total conversion coefficient. In this work, lifetimes ranging from 1 to 100 ps are measured using the plunger technique [START_REF] Dewald | The differential plunger and the differential decay curve method for the analysis of recoil distance doppler-shift data[END_REF].

Experimentally, the ground state spectroscopic quadrupole moment can be measured at zero angular momentum in odd-mass nuclei using laser spectroscopy techniques [START_REF] Campbell | Laser spectroscopy for nuclear structure physics[END_REF]. For excited states, the spectroscopic quadrupole moment can be measured using the Coulomb excitation technique below the Coulomb barrier. At first order, the cross section is proportional to the Rutherford scattering cross section and the probability to populate the final state from the ground state under the electromagnetic field of the nucleus. The cross section can be written such as :

dσ dΩ i→f Coulomb cross section = dσ dΩ Ruth Rutherf ord cross section MiM f | b T ot if | 2 2I i + 1 P i→f probability (1.9)
The b T ot if amplitude has a first order (1) and a second order (2) terms :

b (1) if = 4πZ 1 e i λµ 1 2λ + 1 i | M (Eλ, µ) | f S Eλ,µ b (2)
I f M f IiMi = 1 i 2 n +∞ -∞ I f M f | V (t) | I n M n e iω f n t dt × t -∞ I n M n | V (t ′ ) | I i M i e iωnit ′ dt ′
with S Eλ,µ the Coulomb integrals. The total probabilities is :

P i→f ∝ | b T ot if | 2 ∝| b (2) if | 2 + | b (1) if | 2 +2 | b (1) if b (2) if |
The first order, [START_REF] Heyde | Shape coexistence in atomic nuclei[END_REF], is proportional to the transitional E2 matrix element only while the second order, (2), is proportional to the diagonal E2 matrix element, i.e. the spectroscopic quadrupole moment via the interference term | b

(1) if b (2)
if | [START_REF] Alder | Theory of Coulomb excitation with Heavy ions[END_REF]. The measurement of the Coulomb excitation cross section allows therefore to deduce both reduced transition probabilities and spectroscopic quadrupole moment. Shape coexistence is a particular phenomenon in the study of the nuclear deformation. In that case, a significantly different microscopical configuration than the ground state can give rise to a different shape, corresponding to a second minimum in the potential energy, possibly degenerated with the ground state. The study of this phenomenon remains a challenge for both experimental and theoretical works. The modern challenges are the determination of the shape parameters of the coexisting structures, the coupling between them and how well they can be described by the theories. In the calculations, the main motivation is to understand the origin of the shape coexistence. At the mean-field level, the question is to identify the relevant symmetries. In a more microscopical approach, it is essential to identify the relevant nucleon excitations and coupling between them. Theoretical calculations reached in the last decade a very high level of description of the experimental quantities, in particular based on the Monte Carlo Shell Model approach. The counter part is that the description in proton pair excitation above major shell gap coupled to neutron occupying intruder orbits is not experimentally verified. A long term prospective is therefore the possible direct constrain on this description.

In this manuscript, I have focussed my studies on the shape coexistence phenomena involving the nucleon sub-shell closure 40. The experimental work uses the three experimental techniques described previously: high resolution prompt γ-ray spectroscopy of exotic nuclei to extract the low spin levels scheme, excited states lifetime measurements using the plunger technique and safe Coulomb excitation of radioactive beams. This manuscript is not an introduction to the field. Introductions to the subject, to the experimental techniques and their analysis are available in my PhD manuscript [START_REF] Clément | Shape coexistence in neutron-deficient Krypton and Selenium isotopes studied by low-energy Coulomb excitation of radioactive ion beams[END_REF].

On the contrary, the aim of this work is to summarize very briefly my experimental achievements and put them in the perspectives of more recent experimental works and theoretical calculations. The goal is to identify if the collection of experimental data, on each mass region in which I produced a result, have consistencies and if not, trying to identify them. In chapter 3, controversial experimental results are discussed in great details in the light of the most recent experimental and theoretical works. In all chapters, my personal experimental work are compared and placed in larger systematics including in-beam, β-decay and ground state properties experimental results. The counter part of this choice is that general introductions on each mass region are very brief. Part of the work in chapter 3 are the results of the PhD thesis of A. Dijon [START_REF] Dijon | Evolution of the Collectivity around 68Ni : role of intruder states[END_REF] and I. Celikovic [START_REF] Celikovic | Nuclear structure around doubly-magic nuclei : lifetime measurements in the vicinity of 68Ni and search for isomers around 100Sn[END_REF] at GANIL. In spite my significant involvement in the experimental activities at GANIL through the EX-OGAM and AGATA projects [START_REF] Clément | Conceptual design of the agata 1 03c0 array at ganil[END_REF][START_REF] Akkoyun | Agata advanced {GAmma} tracking array[END_REF], this manuscript will not discussed this aspect of my careerer.

Partie II

Shape coexistence at A=100, Z=40

Chapter 2

Shape coexistence at A = 100, Z=40

Introduction

Neutron-rich nuclei with A∼ 100 are amongst the best examples of interplays between microscopic and macroscopic effects in the nuclear matter. A rapid onset of quadrupole deformation is known to occur at N=60 in the neutron-rich Zr and Sr isotopes, making this region an active area for both experimental and theoretical studies. Already in the 60's, S.A.E. Johansson investigated the properties of light fission fragments of 252 Cf and observed an island of large and constant deformation around A=110 [START_REF] Sven | Gamma de-excitation of fission fragments: (ii). delayed radiation[END_REF]. Since , a large set of experimental data becomes available in this mass region from ground state to high spin excited states.

A compilation of the mass and the charge radii (δ r 2 ) measurements was recently published in [START_REF] Campbell | Laser spectroscopy for nuclear structure physics[END_REF]. In fission fragments with A∼100, the two neutron separation energy shows that the binding energy of Rb, Sr, Y and Zr isotopes rapidly increases at N=60 [START_REF] Audi | The ame2012 atomic mass evaluation[END_REF]. At the same neutron number, the systematics of the δ r 2 shows an abrupt increase of the nucleus radii. Therefore, the onset of stability has been interpreted as a consequence of a dramatic increase of the ground-state deformation. This onset of deformation is unique in the nuclear chart. Indeed, it occurs at exactly N=60 for several elements (Rb, Sr, Y and Zr) ; it is sudden (from spherical N=58 isotones to highly deformed N=60 isotones) ; it is not observed for Mo and Kr isotopes. This low-Z border in the Kr isotopes of this phenomenon has been recently established by means of mass measurements, where no deviation from the standard trend toward the drip line was observed at N=60 [START_REF] Naimi | Critical-point boundary for the nuclear quantum phase transition near a=100 from mass measurements of 96,97kr[END_REF]. This localized effect points to the interaction between specific proton and neutron orbitals.

The systematics of the excitation energy for the first 2 + states in the Sr and Zr isotopic chains show a sudden drop at N=60 (Fig. 2.1) and by applying a simple geometrical model, one can relate it to a change of deformation from β 2 = 0.1 to β 2 = 0.4. On the other hand, in Kr isotopes, the energy of the first 2 + state decreases smoothly between 94 Kr 58 and 100 Kr 68 indicating instead a smooth increase of the deformation [START_REF] Albers | Evidence for a smooth onset of deformation in the neutron-rich kr isotopes[END_REF][START_REF] Flavigny | Shape evolution in neutron-rich krypton isotopes beyond n = 60: First spectroscopy of 98,100 Kr[END_REF]. At N=60, the evolution of the E(4 + )/E(2 + ) ratio between Zr/Sr and Kr shows an abrupt change of trend consistent with a change in microscopical evolution [57].

The uniqueness of this shape change is illustrated in figure 2.2. The systematics of the excitation energy of the 2 + 1 states in neutron rich Sr and Sm isotopes is presented as a function of the neutron number added to the last neutron shell closure, N=50 and N=82 respectively. A maximum of excitation energy is observed at the shell closure and a minimum when 10 neutrons are added. In between, the transition from the spherical shape associated to the shell closure to the highly deformed 2 + 1 is gradual in Sm isotopes where it is sudden in Sr isotopes. Due to the sudden onset of deformation, this shape transition at N=60 and Z∼40 has recently been identified as a Quantum Phase Transition with N as the order parameter [START_REF] Naimi | Critical-point boundary for the nuclear quantum phase transition near a=100 from mass measurements of 96,97kr[END_REF][START_REF] Togashi | Quantum phase transition in the shape of zr isotopes[END_REF].

Finally, low-lying 0 + states, indicating possible shape coexistence, have been identified in the Zr and Sr chains and, similar to the 2 + 1 state, a drop of the 0 + 2 energy is observed at N=60 (Fig. 2.3). A shape coexistence scenario was therefore proposed where the 0 + 2 states for N<60 correspond to a deformed configuration, which then become the ground state at N=60, while the spherical configuration of the ground state for N<60 becomes non-yrast.

The structure of neutron-rich Sr isotopes beyond the first 2 + state has been studied extensively in the past. In 96 Sr, the ground-state band was shown to have a vibrational-like character, and the small B(E2; 2 + 1 → 0 + 1 ) value extracted from the lifetime of 7(4) ps [START_REF] Mach | Retardation of b(e2; 01+ 21+) rates in 9096sr and strong subshell closure effects in the a 100 region[END_REF] is consistent with a nearly spherical ground state. Two low-lying 0 + states at 1229 and 1465 keV were established by G. Jung et al. [START_REF] Jung | Gamma-gamma angular correlations of transitions in sr94 and sr96[END_REF] and interpreted as candidates for a deformed band head, supporting the shape coexistence scenario. An extremely strong electric monopole transition of ρ 2 (E0) = 0.185 [START_REF] Clément | Conceptual design of the agata 1 03c0 array at ganil[END_REF] was observed between these two states [START_REF] Kawade | Excited 0+-levels in98zr[END_REF][START_REF] Lhersonneau | Evolution of deformation in the neutron-rich zr region from excited intruder state to the ground state[END_REF], indicating the presence of a sizeable deformation and strong mixing between the configurations. In 98 Sr, the ground-state band has a rotational character, and the large B(E2) values between the excited states, deduced from lifetime measurements [START_REF] Mach | Deformation and shape coexistence of 0+ states in 98sr and 100zr[END_REF][START_REF] Ohm | The ground-state deformation of 98 sr[END_REF][START_REF] Mach | A method for picosecond lifetime measurements for neutron-rich nuclei: (1) outline of the method[END_REF][START_REF] Lhersonneau | A k=3 two-quasiparticle isomer in 98sr[END_REF][START_REF] Mach | Retardation of b(e2; 01+ 21+) rates in 9096sr and strong subshell closure effects in the a 100 region[END_REF][START_REF] Smith | Spin-dependent triaxial deformation in neutron-rich mo isotopes[END_REF][START_REF] Smith | Lifetime measurements and nuclear deformation in the a≈100 region[END_REF], are consistent with a deformed character of the ground state. A low-lying 0 + 2 state at 215.3 keV was established by F. Schussler et al. [START_REF] Schussler | Discovery of a very low-lying 0+ state in 98sr and shape coexistence implication in 98sr[END_REF] and interpreted as the band head of a presumably spherical structure. A strong electric monopole transition of ρ 2 (E0) = 0.053 [START_REF] Togashi | Quantum phase transition in the shape of zr isotopes[END_REF] was measured between the 0 + 2 and the 0 + 1 states, again supporting the shape coexistence scenario [START_REF] Schussler | Discovery of a very low-lying 0+ state in 98sr and shape coexistence implication in 98sr[END_REF][START_REF] Park | Shape coexistence and evolution in 98 Sr[END_REF].

While these observables yield informations on the configuration of the ground state band and strongly support the shape coexistence scenario, the exact properties of co-existing non-yrast structures and the degree of their mixing with the ground-state configuration can only be inferred from both transition rates and spectroscopic quadrupole moments.

Coulomb excitation experiment results

The spectroscopic quadrupole moments and reduced transition probabilities in 96,98 Sr have been measured by low-energy Coulomb excitation of post-accelerated radioactive ion beams at REX-ISOLDE, which provided firm evidence for shape coexistence and configuration inversion in the neutron-rich Sr isotopes [START_REF] Clément | Spectroscopic quadrupole moments in 96,98 Sr: Evidence for shape coexistence in neutron-rich strontium isotopes at n = 60[END_REF][START_REF] Clément | Low-energy coulomb excitation of 96,98 Sr beams[END_REF][START_REF] Clément | moments in 96,98 Sr: Evidence for shape coexistence in neutron-rich strontium isotopes at n = 60[END_REF]. The corresponding level schemes are presented in figure 2.4.

To summarize the output of this rich set of experimental data :

• A Molecular extraction of a 96 Sr 19 F + beam was developed further broken in the REX-EBIS to deliver a pure secondary 96 Sr beam

• In REX-Trap/REX-EBIS β-decay of 98 Rb + was developed to produce a secondary postaccelerated 98 Sr beam

• 2(9) E2 transitional matrix elements have been measured in 96 Sr ( 98 Sr), respectively.

• 1(5) E2 diagonal matrix elements have been measured in 96 Sr ( 98 Sr), respectively for the first time.

• The B(E2) transition strengths are in good agreement with the results from the most recent lifetime measurements.

• Using the precise lifetimes measurements as additional input in the analysis of the Coulomb excitation data enhanced the sensitivity to the reorientation effect and allowed extracting spectroscopic quadrupole moments for several excited states for the first time.

• In 96 Sr, the diagonal matrix element of the 2 + 1 state is found small and compatible with 0 in the error bars, and compatible with a nearly spherical state. 96 Sr and 98 Sr in [START_REF] Clément | Spectroscopic quadrupole moments in 96,98 Sr: Evidence for shape coexistence in neutron-rich strontium isotopes at n = 60[END_REF][START_REF] Clément | Low-energy coulomb excitation of 96,98 Sr beams[END_REF][START_REF] Clément | moments in 96,98 Sr: Evidence for shape coexistence in neutron-rich strontium isotopes at n = 60[END_REF].

• In 98 Sr, the diagonal matrix elements of the 2 + 1 , 4 + 1 , 6 + 1 and 8 + 1 states are large and negative underlying the large prolate deformation of the ground state band. On the contrary, the 2 + 2 diagonal matrix is found compatible with 0 indicating a spherical state similar to the 2 + 1 in 96 Sr. This is the first ever proof of shape coexistence in Sr isotopes and supports the shape inversion between 96 Sr and 98 Sr.

• The B(E2; 2 + 1 → 0 + 1 ) in 96 Sr is close to the B(E2; 2 + 2 → 0 + 2 ) in 98 Sr supporting their similar nature. In addition, the B(E2;2 + 2 → 0 + 1 ) in 98 Sr are found an order of magnitude smaller than the B(E2; 2 + 2 → 0 + 2 ), suggesting a very weak mixing between the wave function of the coexisting states.

• The systematics of B(E2)'s along the ground state band is found similar in Sr and Zr as good rotational bands.

• They differ from the Mo isotopes which present a change from axial to triaxial rotational band.

• The transitional quadrupole moments calculated from the B(E2,2 + 1 → 0 + 1 ) of 96,94 Kr, 96 Sr, 98 Zr and 100 Mo are found very similar supporting the fact that they lie out of the island of deformation.

• The spectroscopic quadrupole moments of the 2 + 1 state deduced from the diagonal E2 matrix elements in 96,94 Kr, 96,98 Sr and 100 Mo are found very similar and low, suggesting a certain level of triaxility in the deformed 98 Sr, similarly to the deformed 100 Mo.

• The matrix elements were interpreted in a phenomenological two-band mixing model for 98 Sr.

The mixing angle for the 0 + (2 + ) states yield 0.87(1) (0.99 [START_REF] Heyde | Shape coexistence in atomic nuclei[END_REF]) respectively underlying the very weak mixing of the wave function in spite of their proximity in excitation energy.

• The purity of the wave functions increases rapidly with spin, explaining the increase in the spectroscopic quadrupole moments between the different 2 + and 0 + states.

• The quadrupole sum rule formalism was applied to derive shape parameters for the 0 + states.

• The first rotational invariant, Q 2 (similar to |β 2 |), is found small and very similar for the 0 + 1 (0 + 2 ) states in 96 Sr ( 98 Sr) respectively, confirming their weakly deformed character. On the contrary, the value is found large for the 0 + 1 in 98 Sr supporting the large deformation of the ground state band.

• The second rotational invariant, cos3δ (similar to γ) can only be calculated in 98 Sr with the available E2 matrix elements. It shows that both 0 + states are triaxial with a value of γ ∼ 20 • . The similarity of the electromagnetic moments between the 0 + 1 in 96 Sr and 0 + 2 states in 98 Sr suggests that the 0 + 1 in 96 Sr should also present a certain level of triaxility.

• The sharp transition is associated to a very weak mixing between competing configuration, involving triaxility at very low spin, in contrast to the N∼Z cases in Kr and neutron deficient Hg for instance.

• The Davydov-Filippov model was applied on the 2

+ 1 |E2|2 + 1 / 2 + 1 |E2|0 + 1
matrix elements ratio as a function of the γ triaxial parameter. The experimental ratio is compatible with γ ∼ 27 • close to the 20 • extracted from the quadrupole sum rule formalism.

• The transition strengths and quadrupole moments were compared to theoretical calculations going beyond the static mean-field approach using the Gogny D1S interaction which well reproduce the shape coexistence in N=Z nuclei (see Chapter 4).

As mentioned, we applied a simple two-states mixing model to the measured reduced matrix elements in 98 Sr. In this model, the observed physical states |I + 1 and |I + 2 may be expressed as linear combinations of pure prolate and spherical configurations, |I + p and |I + s , respectively:

|I + 1 = + cos θ I × |I + p + sin θ I × |I + s |I + 2 = -sin θ I × |I + p + cos θ I × |I + s
Experimental data, in particular E2 matrix elements, can be used to calculate the mixing amplitudes, cos 2 θ I , between the two pure (unperturbed) configurations. Following the method described [START_REF] Clément | Shape coexistence in neutron-deficient krypton isotopes[END_REF], and using the complete set of E2 matrix elements in 98 Sr we have extracted, small mixing angles for the 0 + and 2 + wave functions [START_REF] Clément | Spectroscopic quadrupole moments in 96,98 Sr: Evidence for shape coexistence in neutron-rich strontium isotopes at n = 60[END_REF][START_REF] Clément | moments in 96,98 Sr: Evidence for shape coexistence in neutron-rich strontium isotopes at n = 60[END_REF][START_REF] Clément | Low-energy coulomb excitation of 96,98 Sr beams[END_REF]. An evaluation of the mixing angles for the 0 + and 2 + states requires a large set of E2 matrix elements which are not available for many nuclei in this mass region. Assuming a negligible mixing between 2 + states, one can extract an approximate mixing angle for the 0 + states via 98 Zr, only an upper limit of the B(E2) is known. The result for 98 Sr is similar to the values determined for neighbouring nuclei. A rather weak mixing between coexisting structures is observed for nuclei in this mass region, both inside and outside the region of deformation. 100 Mo presents a certain level of mixing which could be related to a certain level of triaxiality as proposed from the systematics in B(E2) along the rotational band [START_REF] Wrzosek-Lipska | Electromagnetic properties of 100mo: Experimental results and theoretical description of quadrupole degrees of freedom[END_REF].

tan θ 0 = 0 + 2 |E2|2 + 1 0 + 1 |E2|2 + 1 .

Q s systematics

The diagonal E2 matrix elements extracted from the Coulomb excitation analysis can be translated into spectroscopic quadrupole moments, defined in the laboratory frame, using :

Q s = 16π 5 1 √ 2I + 1 (I, I, 2, 0|I, I) I E2 I ,
It can be compared to the spectroscopic quadrupole moments of the ground states obtained from laser spectroscopy for odd-mass neighbouring nuclei [START_REF] Stone | Table of nuclear electric quadrupole moments[END_REF]. In order to eliminate spin, mass and charge dependences, we further convert the Q s values to intrinsic quadrupole moment and to quadrupole deformation parameters β 2 , assuming an axial symmetry:

Q s = 3K 2 -I(I + 1) (I + 1)(2I + 3) Q 0 = 3K 2 -I(I + 1) (I + 1)(2I + 3) 3 √ 5π ZeR 2 β 2
where R is defined as R=1.

25×A 1/3 [fm].
The comparison for all known Q s is shown in Fig. 2.5 as a function of the neutron number for the odd-neutron Kr, Sr, Zr, Mo and Ru nuclei, and for the odd-proton Rb, Y, Nb and Tc isotopes [START_REF] Stone | Table of nuclear electric quadrupole moments[END_REF]. The spherical neutron Nilsson orbitals are also indicated. The nuclei with N<60 have a low deformation with β 2 ≤0.2. Our measurement of the spectroscopic quadrupole moment for the 2 + 1 state in 96 Sr fits well with the systematics. For N≥60, the deformations of the ground states of odd-mass isotopes, and in the ground state band of 98 Sr, are consistently large, whereas the β 2 value for the 2 + 2 state in 98 Sr is as low as those observed for N<60. Unfortunately, quadrupole moments for 90,92 Rb, 91-93,95-98 Y, 94-98,100 Nb and 99-101 Mo have not been measured yet. The spectroscopic quadrupole moments in 88,90,92 Kr, 90,92,94 Sr and 92-94,96-100 96,98 Sr are compared to those obtained from laser spectroscopy for the ground states of odd-mass nuclei.

States belonging to the ground state band (2 + 2 ) are labelled in red (blue), respectively. Spherical Nilsson orbits are mentioned for neutron. stable 94,96 Mo and 98,100,102 Ru nuclei the quadrupole moments of the 2 + 1 states were measured using the reorientation effect with light-ion beams and yielded two alternative solutions in the Coulomb excitation analysis leading to different conclusions on the deformation. The data are therefore not displayed in the figures. This limitation could be overcome using multi-step Coulomb excitation with high-Z collision partners. The systematics shown in Fig. 2.5 is further extended by including the spectroscopic quadrupole moments of the 2 + 1 states in 94,96 Kr [55, 71], 98,100 Mo [72, 69] and 104 Ru [73], extracted from low-energy Coulomb excitation experiments. They follow consistently the trend observed in laser spectroscopy. Between N=50 and 58, when the d 5/2 and s 1/2 orbits are filled, the deformation is uniformly low ( β 2 ∼ 0.1) for all elements from Kr to Tc. From N=60, all deformation parameters are uniformly high ( β 2 ∼ 0.5) for all elements from Rb to Tc. Due to the large error bars, no conclusion can be drawn for 96 Kr. Unfortunately, 102 Mo is not yet measured as possible high even-Z border of the island of deformation. The onset of deformation when adding 2 neutrons to N=58 (ie. the s 1/2 orbit) to N=60 (ie filling the g 7/2 ) is consistently observed in both methods. This systematics highlights a particular behaviour at N=59. As the onset of deformation is abrupt for all elements from Z=37 to 41 when adding a pair of neutrons, at N=59 a gradual effect as function of the proton number emerges. The 96 37 Rb 59 deformation is estimated at β 2 ∼ 0.17, slightly higher than the N≤58 systematics and well below the N≥60 systematics. For 103 44 Ru 59 , the deformation is evaluated at β 2 ∼ 0.4, slightly lower than the N≥60 systematics. Therefore, a more gradual shape change is suggested when only one neutron is added to the g 7/2 and when increasing the proton number in the pf shells. The shape change seems highly correlated to the occupancy of both proton and neutron in the mentioned orbits. Unfortunately, this gradual change between Sr and Tc cannot be quantified as no laser spectroscopy of this N=59 isotone (1/2 ground state spin) was performed so far. In figure 2.6, the deformation is represented in the (N,Z) plan. The sizes of the squares in the plot are proportional to β 2 . These systematics suffers from several missing experimental points. Large quadrupole deformations are observed for the ground states of odd-mass nuclei and for the first 2 + states in even-even nuclei for elements between Z=37 and Z=44 at N=60 and beyond. With this two dimensional systematics, the island of deformation appears clearly and shows its boundaries. It appears that the Kr isotopes lie outside the island of deformation since the spec-troscopic quadrupole moment in 96 Kr 60 is low with respect to those in 97 Rb 60 and 98 Sr 60 . This may indicate that for Z=36, the two neutrons beyond N=58 occupying the g 7/2 are insufficient to induce a large quadrupole deformation. The completion of the systematics, in particular for the Kr isotopic chain and N=59 isotones, are vital informations to further constrain the microscopic interpretation of the onset of deformation at N=60. In figure 2.7, the algebraic deformation is deduced (β 2 ). The blue (green) squares indicate prolate (oblate) deformations respectively. The first matrix shows the deformation for the ground states in odd-nucleon isotopes and the 2 + 1 state for even-even isotopes. The second matrix displays the long lived isomeric state deformation in odd-proton or odd-neutron isotopes extracted from the laser spectroscopy. In the case of the even-even nuclei, the 2 + 2 or 2 + 3 state built on top of the 0 + 2 state with the largest B(E2) is shown. The first systematics is similar to figure 2.6 and highlights the onset of deformation of prolate character. The second systematics reveals the lack of experimental data in spite of the large set of known long lived isomers. Unfortunately, only spectroscopic quadrupole moment in Y isotopes have been measured. In even-even nuclei, 98,100 Mo and 98 Sr have measured spectroscopic quadrupole moments. Consistently, for isotopes with N≤58, non-yrast 2 + states and isomers have very large deformation of prolate character in Mo and oblate character in Y isotopes coexisting with weakly deformed ground state. Beyond N=58, only one deformation in excited configuration is measured in 98 Sr having a weak oblate deformation. This systematics shows again the consistency between the laser spectroscopy measurement and the Coulomb excitation data and the shape inversion at N=60. Below Z=40, the non-yrast configuration seems to be systematically of oblate character. Spectroscopy of fission fragments with mass A∼ 100 produced in fission sources or induced fission of actinide has triggered a large experimental effort world-wide to probe the scenario of shape coexistence. Large set of γ-ray spectroscopy and electron spectroscopy at high spins are available in the literature. The onset of collectivity and deformation change in the ground state is now clearly established, even if not for all isotopic chains. On the contrary, the shape coexistence description is incomplete and systematics of excited and ground state deformation are very scarce. With the development of post-accelerated radioactive ion beams, experimental data with "direct" methods are available and improve our description of shape coexistence in this mass region. The spin and parity of the ground state and long-lived isomers give strong indication on the involved Nilsson orbits. Figure 2.8 shows the systematics of measured ground states and isomeric states spin/parity and highlight confirmed deformation in Zr, Y and Sr isotopes as a function of the neutron number. Isotones are presented in the same column and a colour code indicates the element. For clarity, figure 2.9 shows the orbitals involved in the present discussion. , presented with a dashed line is adopted from the calculations in [START_REF] Togashi | Quantum phase transition in the shape of zr isotopes[END_REF]. The orbitals energy spacing is arbitrary. The microscopical configuration of the spherical ground state of 96 Sr is shown.

Island of inversion

Overview with other experimental data 2.4.1 Bands head overview

Even neutron number

For even neutron number, the systematics is shown for ytrium (Z=39). Using spherical Nilsson orbits, the single proton occupies the 2p 1/2 orbit as proven by the confirmed J π = 1/2 -ground state assignment up to neutron number N=58. The onset of deformation is triggered by proton-particle hole excitation across Z=40. The confirmed J π = 5/2 + ground state for 99,101 Y involves either a single proton into 2d 5/2 lying above the g 9/2 orbits or a strongly coupled deformed even-even core and a single proton into the [422]5/2 + Nilsson orbital (prolate deformed 1g 9/2 ). This second configuration is favoured from both the laser spectroscopy [START_REF] Cheal | The shape transition in the neutron-rich yttrium isotopes and isomers[END_REF] and in-beam γ-rays spectroscopy [START_REF] Monnand | Evidence for a rotational band in99y[END_REF][START_REF] Wohn | Rotational structure and nilsson orbitals for highly deformed odd-a nuclei in the a ∼ 100 region[END_REF][START_REF] Hagen | Evolution of nuclear shapes in odd-mass yttrium and niobium isotopes from lifetime measurements following fission reactions[END_REF] with a deformation β ≃0.4.

In spherical ytrium isotopes, very long lived isomers have been observed. From laser spectroscopy, a spin/assignment J π = 9/2 + was deduced together with a large spectroscopic quadrupole moment indicating a large oblate deformation [START_REF] Stone | Table of nuclear electric quadrupole moments[END_REF]. This is compatible with a single proton excited into the [404]9/2 + Nilsson orbital (oblate deformed 1g 9/2 ). In the deformed configuration, one can expect a large fragmentation of the neutron occupancy in the 1νg 7/2 and 1νh 11/2 orbits.

Beyond N=60, the ground states are highly deformed and no isomeric state has been reported so far.

Even proton number

For even proton number, the systematics is shown for strontium (Z=38) and zirconium (Z=40). In figure 2.8, both at N=57 and N=59, Sr have a confirmed J π = 1/2 + ground state. This spin assignment in the supposed spherical 97 Zr and 95 Sr (N=57) ground state and is compatible with a single neutron in the 3s 1/2 spherical Nilsson orbit. This assumption is supported by the local maximum of the 2 + 1 excitation energy in 96,98 Zr in figure 2.1 corresponding to two neutrons filling the 3s 1/2 spherical Nilsson orbit. With one neutron more in 97 Sr (N=59), this spin/parity assignment to J π = 1/2 + is surprising. At N=59, the 3s 1/2 is full and the ν1g 7/2 starts to be filled with one neutron in the conventional spherical Nilsson orbit. In a spherical framework, the 3s 1/2 orbit is the only available low energy J=1/2 level for N=59 nucleus. As mentioned earlier from the proposed Q s systematics, the N=59 appears as a transitional region and could present a certain level of deformation. According to MCSM calculations [START_REF] Togashi | Quantum phase transition in the shape of zr isotopes[END_REF], in Zr isotopes, the neutron 2d 3/2 orbit is placed between the 3s 1/2 and the 1g 7/2 and could provide a J π = 1/2 + spin/parity when deformed (1/2[400]). Therefore, the possible weakly deformed ground state at N=59 must be based on a mixture of weakly deformed s 1/2 , d 5/2 , d 3/2 and g 7/2 neutron orbits which differ from the large deformation of the ground state beyond N=60.

Isomeric states have drawn a particular attention below N=60 as possible candiate for shape isomers [START_REF] Urban | Medium-spin structure of 96,97sr and 98,99zr nuclei and the onset of deformation in the a100 region[END_REF][START_REF] Urban | The ν9/2[404] orbital and the deformation in the a 100 region[END_REF][START_REF] Stone | Table of nuclear electric quadrupole moments[END_REF]. Long-lived 9/2 +m states have been identified in several odd Sr and Zr isoropes but unfortunately no spectroscopic quadrupole moment were measured so far. However, rotational bands have been clearly established supporting the deformed character, and therefore shape coexistence, in these isotopes. It is proposed in [START_REF] Urban | The ν9/2[404] orbital and the deformation in the a 100 region[END_REF] that the 9/2 +m isomeric state involves a neutron excitation from the deformed 9/2[404] (g 9/2 )orbit from the 78 Ni core into the deformed 3/2[541] (h 11/2 ).

In the deformed Nilsson diagram for excited configurations, at N = 59, the odd neutron can be placed either on the 3/2[541] (h 11/2 ) orbital, making a 3/2 -rotational band, or the 3/2[411] (g 7/2 ) orbital, making a 3/2 + rotational band. The deformation is larger than at N = 58 but the 9/2[404](g 9/2 ) orbital still limits the deformation. The deformed 3/2 + and 3/2 -excited band heads have been observed experimentally (not shown in the figure, see [START_REF] Urban | Medium-spin structure of 96,97sr and 98,99zr nuclei and the onset of deformation in the a100 region[END_REF]) and are based on the single neutron occupancy of these deformed orbits.

In the deformed region, at N=61 and 63, laser spectroscopy have established the deformed (prolate) character of 101 Zr and 101 Sr ground states. Tentative spins and parities are (3/2 + ) and (5/2 -) respectively. At N = 61 the situation is analogous to that at N = 59. Two additional deformed bands with the single neutron in the 3/2[411] (g 7/2 ) and 5/2[532] (h 11/2 ) orbitals are observed. It is worth mentioning that the 9/2 +m K-isomer rotational band (similar to N=59) still exists at N=61 in the deformed 101 Zr. It means that it is not the configuration based on the ν9/2[404] excitation which becomes the ground state at N=60. In other words, the deformed ground state beyond N=60 does not involve, from the experimental point of view, an excitation from the 78 Ni core. As described later in the section 2.5.2, Monte Carlo Shell Model Calculations lead to the same conclusion and do not involve any neutrons excitation from the 78 Ni core to reproduce the large deformation beyond N=60.

Transition probabilities overview

Numerous γ-rays experimental data set exist probing the onset of deformation, from level schemes to B(E2) values by lifetime measurement, which give more indirect proof of the shape and microscopical configurations compared to Coulomb excitation experiments. A list of recent relevant works in the last 5 years in which I have been involved is listed below in addition to the Coulomb excitation data presented before.

• B(E2;2 + 1 → 0 + 1 ) value in 90 Kr [START_REF] Régis | b(e2; 2 + 1 → 0 + 1 ) value in 90[END_REF].

• Abrupt shape transition at neutron number N=60: B(E2) values in 94,96,98 Sr from fast γ timing, [START_REF] Régis | Abrupt shape transition at neutron number n = 60: b(e2) values in 94,96,98 Sr from fast γ -γ timing[END_REF].

• Evolution of nuclear shapes in odd-mass yttrium and niobium isotopes from lifetime measurements following fission reactions, [START_REF] Hagen | Evolution of nuclear shapes in odd-mass yttrium and niobium isotopes from lifetime measurements following fission reactions[END_REF].

• Experimental study of the lifetime and phase transition in neutron-rich 98,100,102 Zr, [START_REF] Ansari | Experimental study of the lifetime and phase transition in neutron-rich 98,100,102 Zr[END_REF].

• Evidence for coexisting shapes through lifetime measurements in 98 Zr, [START_REF] Singh | Evidence for coexisting shapes through lifetime measurements in 98 Zr[END_REF] As a result, the B(E2;2 + 1 → 0 + 1 ) systematics is now completed from N=50 to N=60, ie covering from the neutron shell closure to the shape transition, for Zr, Sr and Kr isotopes. The systematics is presented in figure 2.10 where the B(E2) in W.u. are normalized by the mass A for each isotopes. This representation first proposed by R. F. Casten and N. V. Zamfir [START_REF] Casten | Valence correlation schemes and signatures of nuclear structure: A simple global phenomenology for b(e2:2 + 1 →0 + 1 ) values[END_REF] illustrates the amount of nucleons participating to the collectivity. If the ratio equals unity, it illustrates that all nucleons participate to the collective behaviour. In a nearly spherical nucleus, this ratio should be close to 0 whereas in highly collective nucleus it should approach 1. In the case of the Sr/Zr isotopes, the onset of collectivity follows the two extremes. Below N=60, the ratio is very close to 0. For 96,98 Zr, the local maximum of the 2 + 1 excitation energy (see figure 2.1), reflecting the νs 1/2 occupancy, corresponds to a local minimum in figure 2.10. From N=60, the ratio saturates where all nucleons participate to the collective behaviour suggesting an almost perf ect rigid rotor.

Rotational band and configuration mixing

From the collective model of A. Bohr and B. R. Mottelson, in a rotational band, the spectroscopic quadrupole moment, ie the intrinsic moment of the 2 + 1 state, equals the transitional quadrupole moment that one can convert from the B(E2;2 + 1 → 0 + 1 ) as follow :

Q s (I, K) = 3K 2 -I(I + 1) (I + 1)(2I + 3) • Q 0
where Q 0 is the intrinsic quadrupole moment of the I state (ie in the nucleus frame) and Q s is the spectroscopic quadrupole moment (ie in the laboratory frame). I is the angular momentum with its projection K. In the rotational model

B(E2; (I i , K) → (I f , K)) = (5/16π) • Q 2 0 • (I i K20|I f K) 2
.

For an even-even rotational nucleus and for its first 2 + , I = 2, K = 0, the rotor model gives :

Q s (2, 0) = -(2/7) • Q 0 .
Simplifying by Q 0 , the asymptote limit is

Q s (2, 0) B(E2; (0, 0) → (2, 0)) = -0.906
In [START_REF] Sharon | Systematics of the ratio of electric quadrupole moments q(21+) to the square root of the reduced transition probabilities b(e2;01+??21+) in even??even nuclei[END_REF], a global survey of the 2 + 1 states collectivity is proposed using a quantity, R QB , defined as :

R QB = - Q s (2 + 1 ) B(E2; 0 + 1 → 2 + 1 )
In rare-earth elements, known to be excellent rotor, the E x (4 + 1 )/E x (2 + 1 ) ratio saturates at 10/3 [START_REF] Casten | Valence correlation schemes and signatures of nuclear structure: A simple global phenomenology for b(e2:2 + 1 →0 + 1 ) values[END_REF] with the R QB ratio nearly equal to unity. The E x (4 + 1 )/E x (2 + 1 ) ratio equals 1.2 (3.0) in 96 Sr ( 98 Sr) respectively. The B(E2;2 + 1 → 0 + 1 )/A ratio yields a value of 0.18 (0.98) in 96 Sr ( 98 Sr) respectively. This indicates that 96 Sr behaves as a spherical vibrator and 98 Sr as an almost perfect rotor. From our Coulomb excitation data, spectroscopic quadrupole moments for the 2 + 1 state in 96,98 Sr were measured at -22(30) ef m 2 and -52 [START_REF] Lhersonneau | Evolution of deformation in the neutron-rich zr region from excited intruder state to the ground state[END_REF] ef m 2 respectively. The accurate value of the B(E2), leads to R QB ( 96 Sr) = 0.46 [START_REF] Stone | Table of nuclear electric quadrupole moments[END_REF] and R QB ( 98 Sr) = 0.45 [START_REF] Kelsall | [END_REF]. The large error bars on the experimental Q s do not allow drawing final conclusion for 96 Sr. However, within the experimental error bars, the value for 98 Sr is found far from what one expect for a good rotational nucleus.

As mentioned earlier [START_REF] Clément | Spectroscopic quadrupole moments in 96,98 Sr: Evidence for shape coexistence in neutron-rich strontium isotopes at n = 60[END_REF][START_REF] Clément | moments in 96,98 Sr: Evidence for shape coexistence in neutron-rich strontium isotopes at n = 60[END_REF][START_REF] Clément | Low-energy coulomb excitation of 96,98 Sr beams[END_REF], using the full set of E2 matrix elements, the two-level mixing model was applied in 98 Sr. A mixing angle of 0.87(1) was extracted for the 0 + 1 state wave function while the 2 + states have almost no mixing. U nperturbated matrix elements can be extracted from which an unperturbated intrinsic quadrupole moment for the band can be deduced, later translated in the laboratory frame. The unperturbated matrix elements equals -1.45( 2) eb (to be compared to the measured -0.63( +0. 32 -0.28 ) eb for the 2 + 1 state). This leads to an unperturbated 2), ie in the rotational limit. As a summary, the weak mixing of the 0 + states wave functions does not influence significantly the rotational behaviour of the excitation energies and B(E2).However, the measured spectroscopic quadrupole moment for the 2 + 1 state shows a clear perturbation from its rotational behaviour. Using the two-levels mixing model, the unperturbated moment fits well the rotational behaviour. One must underline that Q un s doesn't involve the measured Q s (2 + 1 ) (see [START_REF] Clément | Shape coexistence in neutron-deficient krypton isotopes[END_REF] for the details ). An approximate experimental Q un s can be estimated from the average measured Q un s in the 4 + , 6 + and 8 + states and equals to -1.6(3) eb [START_REF] Clément | Low-energy coulomb excitation of 96,98 Sr beams[END_REF]. In that case, R Qs QB ( 98 Sr) = 1.4(5), in agreement with the previous observations. As a results, the presented study for 98 Sr shows the influence of the mixing in the shape coexistence and how it can be taken into account in collective nuclei. However, the spectroscopic quadrupole moment of the 2 + 1 state is not discussed here and not used in the present R QB calculations. Its value is found smaller than the Q un s calculated from mixing model and the average value at higher spins. The difference could be attributed to the mixing of the wave functions. However, from the mixing model, it is found negligible. Therefore, the low value is not attributed to the mixing but to a certain level of triaxility as deduced from the sum rules.

Q un s = -110(3) ef m 2 . Using Q un s in the R QB equation leads to R un QB ( 98 Sr) = 0.96(

Particle-Core coupling scheme

In the deformed region, one can also underline that lifetime measurements in odd-proton and even-neutron [START_REF] Hagen | Evolution of nuclear shapes in odd-mass yttrium and niobium isotopes from lifetime measurements following fission reactions[END_REF] illustrate very well the particle+rotor model (so-called strong coupling between a deformed even-even core and a single particle) [START_REF] Shalit | Nuclear Shell Theory[END_REF]. In 99 Y, the ground state spin/parity is 5/2 + . In the particle+rotor model, the ground state can be seen as 5/2 + = π[422]5/2 + ⊗ 0 + ( 98 Sr). The proposed 9/2 + state at 284 keV in 99 Y can be part of the multiplet of the π[422]5/2 + ⊗ 2 + ( 98 Sr) configuration. The B(E2; 2 + → 0 + ) established at 0.259(8) e 2 b 2 in 98 Sr [START_REF] Clément | Spectroscopic quadrupole moments in 96,98 Sr: Evidence for shape coexistence in neutron-rich strontium isotopes at n = 60[END_REF] is compatible with the measured B(E2; 9/2 + → 5/2 + ) = 0.25(2)e 2 b 2 in 99 Y by [START_REF] Hagen | Evolution of nuclear shapes in odd-mass yttrium and niobium isotopes from lifetime measurements following fission reactions[END_REF], supporting very well the particle+rotor model. At N=58, in spherical 96 Sr, the 2 + state excitation energy is 815 keV with a B(E2; 2 + → 0 + ) = 0.045(10) e 2 b 2 . In the weak coupling scheme, 97 Y should present a multiplet of states (1/2 -)⊗2 + ( 96 Sr) = (5/2 -, 3/2 -) with a barycentre equal to the excitation energy of the 2 + state in 96 Sr and similar B(E2) to the ground state. A state at 953 keV with proposed spin/parity assignment (5/2 -, 3/2 -) and a lifetime T 1/2 ≤ 4 ps is suggested. A more accurate measurement of this lifetime would confirm the spin/parity assignment of the ground state and the weak coupling scheme. In lighter isotopes at N=56, the 2 + state excitation energy in 94 Sr is 837 keV with a corresponding B(E2) = 0.020(8) e 2 b 2 . In 95 Y, two states with confirmed spin/parity at 827 keV (5/2 -) and 685 keV (3/2 -) are the obvious candidates for the (1/2 -)⊗2 + ( 94 Sr) multiplet. Unfortunately, the corresponding B(E2) are not measured. If such measurements could confirm the weak coupling scheme, it would bring further indications that the Z=40 is a robust su-shell closure. Indeed such weak-coupling states have been established around large proton shell gap at Z=28, Z=50 and Z=82.

2.4.5 Experimental multi-messenger approach: the 96 Sr case around a weak deformation [START_REF] Clément | Spectroscopic quadrupole moments in 96,98 Sr: Evidence for shape coexistence in neutron-rich strontium isotopes at n = 60[END_REF]. As already mentioned, two 0 + excited states have been observed at 1229 and 1465 keV. E0 spectroscopy was performed and deduced one of the highest ρ 2 (E0) value between the 0 + 2 and 0 + 3 states, indicating a very large difference in deformation and/or large mixing of the wave functions [START_REF] Lhersonneau | Evolution of deformation in the neutron-rich zr region from excited intruder state to the ground state[END_REF]. The E0 branch to the ground state was never established. The 2 + 2 and 2 + 3 states are very close in energy to the 0 + 2 and 0 + 3 states respectively, suggesting a rotational band. The second 4 + , 6 + and 8 + states develop a rotational band but it is very unclear to which band head it belongs. The lifetime of the second 6 + and 8 + states was measured and suggest a rather weak deformation [START_REF] Urban | Medium-spin structure of 96,97sr and 98,99zr nuclei and the onset of deformation in the a100 region[END_REF]. The 95 Sr ground state is a 1/2 + state compatible with the spherical Nilsson orbit ν[s1/2] [86]. Unfortunately, its spectroscopic quadrupole moment was not measured. The neutron transfer reaction 95 Sr(d,p) 96 Sr was recently performed [START_REF] Cruz | Shape coexistence and mixing of low-lying 0+ states in 96sr[END_REF][START_REF] Cruz | Single-particle structure of neutron-rich sr isotopes via 2 H( 94,95,96 Sr, p) reactions[END_REF], which favours spherical component of the populated states are favoured in 96 Sr. It turned out that the ground state is relatively weakly populated as the main spherical strenght is found in the low lying 0 + states, suggesting a rather weak deformation of the ground state (compatible with the spectroscopic quadrupole moment extracted from the Coulomb excitation) and a nearly spherical 0 + 2 state which however presents a large mixing with the deformed 0 + 3 state [START_REF] Cruz | Shape coexistence and mixing of low-lying 0+ states in 96sr[END_REF][START_REF] Cruz | Single-particle structure of neutron-rich sr isotopes via 2 H( 94,95,96 Sr, p) reactions[END_REF]. From the E0 spectroscopy and the large ρ 2 (E0)(0 + 3 → 0 + 2 ), the 0 + 3 becomes a natural candidate for the band head of the deformed configuration that becomes the ground state in 98 Sr. Such measurement was not performed from 97 Zr to 98 Zr having also two excited 0 + states. One could assume a similar scenario and draw a direct comparison between the 0 + 1,2,3 states in 96 Sr and 98 Zr.

However, different conclusions were drawn in the N=58 98 Zr by [START_REF] Singh | Evidence for coexisting shapes through lifetime measurements in 98 Zr[END_REF]. In this work, the B(E2) values in 98 Zr have been measured and compared to MCSM calculations. The calculations propose the coexistence of three different structures at low spin: a nearly spherical ground state (0 + 1 ), a deformed excited 0 + 2 state, and a well-deformed (band-like) structure possibly based on the 0 + 3 state. The calculated proton and neutron occupancies are very similar in both 0 + 2 and 0 + 3 deformed states. Only small differences in the neutron occupancies make the change in deformation. The 95 Sr(d,p) 96 Sr transfer reaction cross sections propose instead nearly spherical 0 + 1 and 0 + 2 states with a well-deformed 0 + 3 state. The angular distributions treated as a single-step process to the 0 + 2 is best fitted for a l = 0 angular momentum transfer, thus neutron populating the ν2s 1/2 orbital. According to the previous discussions, a deformed configuration would involve a l = 2 (d 3/2 ), or l = 4 (g 7/2 ) or l = 5 (h 11/2 ) angular momentum transfer. The microscopical configuration of the 0 + 2,3 state at N=58 remains therefore puzzling.

2.5 Theoretical calculations for 96,98 Sr

Mean field approach

The sharp transition and magnitude of the deformation at A∼100 remain a challenge for theories.

Similarly to the experimental effort, numerous theoretical works have been published (more than 100 theoretical papers since the 70's) on the subject. All nuclear models have been used to reproduce this onset of deformation such as the HFB + the generator coordinate method (GCM), the macroscopic-microscopic method, the shell model, the Monte Carlo shell model, the interacting boson model (IBM) approximation, the VAMPIR model and covariant density functional (DF) theory (PC-PK1). The global comparison of all the models is out of the scope of the present discussion.

In the following, the state-of-the-art of mean-field calculations using the Gogny D1S interaction with GCM-GOA and Monte Carlo Shell Model Calculations will be briefly discussed. We compared the set of E2 matrix elements with calculations using the Gogny D1S interaction with GCM-GOA. The comparison is presented in [START_REF] Clément | Spectroscopic quadrupole moments in 96,98 Sr: Evidence for shape coexistence in neutron-rich strontium isotopes at n = 60[END_REF][START_REF] Clément | moments in 96,98 Sr: Evidence for shape coexistence in neutron-rich strontium isotopes at n = 60[END_REF][START_REF] Clément | Low-energy coulomb excitation of 96,98 Sr beams[END_REF]. Bands assignment in calculated states is not obvious. We developed a graphical method detailed in [START_REF] Clément | Low-energy coulomb excitation of 96,98 Sr beams[END_REF] to construct theoretical bands. In figure 2.12, the calculated level scheme is compared to the experimental one for 96 Sr. As discussed in [START_REF] Clément | Spectroscopic quadrupole moments in 96,98 Sr: Evidence for shape coexistence in neutron-rich strontium isotopes at n = 60[END_REF][START_REF] Clément | Low-energy coulomb excitation of 96,98 Sr beams[END_REF], the Gogny calculations over estimate the collectivity at N=58 and under estimate the deformation at N=60. The calculated ground state band differs from the experimental ones in B(E2) and predict a very fast increase of the Q s as a function of spin with a significant contribution of K=2 (30%) for the 2 + 1 state leading to this low Q s . This is impossible to verify today. The calculations do not predict two 0 + excited states in contrast to the experimental situation. The (B) band presents large B(E2) associated to low Q s with only 50% of K=0 underlying the triaxial character of the state. In the (A) band, large B(E2)'s are associated to large Q s with a large contribution of K=0. Figure 2.13 shows the comparison between the Gogny calculations and the VAMPIR approach [START_REF] Petrovici | Triple shape coexistence and shape evolution in the n = 58 sr and zr isotopes[END_REF]. The VAMPIR B(E2) are mentioned in red. The agreement is surprisingly good in spite of the very different approaches used in the calculations. Figure 2.13: Comparison between the experimental and calculated level scheme of 96 Sr using the Gogny D1S interaction and the VAMPIR approach In figure 2.14, the calculated level scheme is compared to the experimental one for 98 Sr using the Gogny interaction. In both cases, the ground state band has a large prolate deformation. It is worth mentioning that in spite of the differences in absolute values, the Gogny calculations predict a drop of the Q s between the 4 + 1 and the 2 + 1 states as observed experimentally. A K=2 γ-band is predicted very close to the ground state band but not yet observed experimentally. The nearly spherical configuration is assigned as the (A) band based on the similarities in the B(E2) connecting to the ground state band.

A summary of the experimental ρ 2 (E0) transition with the calculated ones using Gogny D1S, Excited VAMPIR and 5DCH (PC-PK1) in table 2.2. It must be underlined that discrepancies are huge and prediction not always calculated for all transitions making the comparison difficult.

Shell model approach

In the Sr and Zr isotopic chains, the spherical-to-deformed transition takes place when going from 58 to 60 neutrons, thus when the νg 7/2 orbital is being filled. Shell model calculations were performed for the Zr isotopic chain in an extended model space [START_REF] Sieja | Shell model description of zirconium isotopes[END_REF][START_REF] Togashi | Quantum phase transition in the shape of zr isotopes[END_REF][START_REF] Togashi | Quantum phase transition in the shape of zr isotopes[END_REF][START_REF] Singh | Evidence for coexisting shapes through lifetime measurements in 98 Zr[END_REF] and pointed to the π -ν interaction between the spin-orbit partners π0g 9/2 and ν0g 7/2 as the main mechanism for the shape change: as the ν0g 7/2 orbital is being filled, the Z=40 sub-shell gap between the π0f 5/2 and π0g 9/2 effective single-particle energies (ESPE) is reduced, giving rise to multiple particle-hole excitations Table 2.2: Monopole transition strengths ρ 2 (E0) between 0 + states. Experimental results are compared with those calculated within the beyond-mean-field approach with a five-dimensional collective Hamiltonian and the Gogny D1S force (present study) and using the complex excited VAMPIR approach [START_REF] Petrovici | Triple shape coexistence and shape evolution in the n = 58 sr and zr isotopes[END_REF]. Figure 2.14: Comparison between the experimental and calculated level scheme of 98 Sr using the Gogny D1S interaction across the gap. The most recent calculations [START_REF] Togashi | Quantum phase transition in the shape of zr isotopes[END_REF][START_REF] Togashi | Quantum phase transition in the shape of zr isotopes[END_REF][START_REF] Singh | Evidence for coexisting shapes through lifetime measurements in 98 Zr[END_REF] using MCSMC show a spectacular agreement with the experimental data in even-even nuclei and become the reference in the mass region. In the calculations [START_REF] Kremer | First measurement of collectivity of coexisting shapes based on type ii shell evolution: The case of 96zr[END_REF][START_REF] Singh | Evidence for coexisting shapes through lifetime measurements in 98 Zr[END_REF], the 0 + 2 and 0 + 3 below N=60 result from 2p-2h proton excitations (with a possible 4p-4h contribution) from the pf -shell into the π0g 9/2 orbital. On the neutron side, the νg 9/2 remains fully filled, ie without excitation from the 78 Ni core. The deformed configurations in even-even excited 0 + does not involve neutron excitation from the N=50 closed shell. This might explain why (9/2 +m ) deformed states still remains beyond N=60 in odd-neutron Sr/Zr isotopes as excited configurations. In the deformed configuration, neutrons are well distributed from the d 5/2 into the s 1/2 ,d 3/2 and more significantly into the g 7/2 and h 11/2 as deduced from the previous studies of odd neutron isotopes. A similar mechanism is known to be responsible for the rapid onset of deformation and shape coexistence in neutron-rich isotopes around N= 8, 20, 28 and 40, with different spin-orbit partners [91, 92, 1, 93, 94].

Conclusions

For the future, several systematics are needed :

• 98 Sr remains a unique case in term of experimental data. More are needed in particular in the N=58 and N=59 to understand the shape evolution along the yrast bands and excited deformed configuration.

• Coulomb excitation of 96 Sr, Zr and Kr isotopes are mandatory to determine the shape of the different configuration.

• Following the exploratory work on the 95 Sr(d,p) reaction, single and double nucleon transfer experiment are needed to probe both the proton and neutron microscopic configuration.

• The N=59 isotonic line deserves a particular interest in laser and direct nucleon transfer experiment.

• In the spherical isotopes, the weak coupling scheme was not yet studied in details.

• The differences in structure and deformation of the 0 + 2 and 0 + 3 below N=60 in even-even Sr and Zr isotopes must be clarified.

• Laser spectroscopy of long lived isomeric state from N=50 to N=60 must be pursued as well as for ground state beyond N=60.

• In the global survey of the spectroscopic quadrupole moment, the safe Coulomb excitation of 102 Mo will make the bridge between the island of large deformation and the triaxial shape region

• Regarding the very large amount of experimental data in the mass region, holistic comparison in theoretical work must be encouraged.

Partie III

Shape coexistence in the neutron rich Ni isotopes Chapter 3

Shape coexistence at N=40 and Z=28

Introduction

In isotopes with Z<28 and N=40, a new region of deformation was identified and characterized in the recent years. When removing only two protons from 68 Ni to 66 Fe, first excited states and reduced transition probabilities have indicated an onset of collectivity. This mass region presents a difference with the two other cases presented in the manuscript. The Z=28 shell closure is a strong shell closure as revealed by the doubly magic character of 56 Ni [START_REF] Yurkewicz | Nuclear structure in the vicinity of n = z = 28 56 Ni[END_REF] and 78 Ni [START_REF] Xu | β-decay half-lives of 76,77 Co, 79,80 Ni, and 81 Cu: Experimental indication of a doubly magic 78[END_REF][START_REF] Taniuchi | 78ni revealed as a doubly magic stronghold against nuclear deformation[END_REF] by opposition to the Z=40 sub-shell closure. The corresponding large proton gap implies that, in the scheme of the π particle-hole excitation as standard mechanism leading to shape coexistence, it must involve a substantial gain in π -ν quadrupole correlation energy. Along the Ni isotopic chain, neutrons are added to the νg 9/2 orbital from 69 Ni to the doubly magic 78 Ni. The experimental level schemes for the yrast band in even-even nuclei show a 8 + isomeric state arising from the alignment of a broken neutron pair as an example of seniority ν = 2 scheme. The resulting multiplet should be pure neutron states, ie with very low E2 probabilities. In 70 Ni, the 8 + and 6 + states present such character as the enhanced B(E2) for the 2 + state was interpreted as an indication that the Z=28 proton shell gap, near 68 Ni, is relatively weak leading to core polarization [START_REF] Grzywacz | New island of µs isomers in neutron-rich nuclei around the shell closures[END_REF][START_REF] Perru | Enhanced core polarization in 70 Ni and 74 Zn[END_REF][START_REF] Lisetskiy | New t = 1 effective interactions for the fpg model space: Implications for valence-mirror symmetry and seniority isomers[END_REF].

From the B(E2) measurements in Ni, it was proposed that the spin-orbit splitting πf 7/2 -f 5/2 is reduced when filling the νg 9/2 orbital through the tensor term of the nuclear interaction favouring the development of collectivity and possibly shape coexistence. Already in the 80's, M. Girod et al [START_REF] Girod | Spectroscopy of neutron-rich nickel isotopes: Experimental results and microscopic interpretation[END_REF] predicted a low lying deformed 0 +m state in 68 Ni as a first prediction of shape coexistence at N=40. Deformed states below N=40 would therefore involved the νg 9/2 as intruder orbit. 68 Ni is therefore cornerstone of the mass region. The proton shell gap is closed at Z=28 by the f 7/2 orbits and the next shells are the p 3/2 , f 5/2 and p 1/2 . On the neutron side, the N=40 gap is defined between the p 1/2 and the g 9/2 , d 3/2 orbitals. Experimentally, at N=40, the mass measurements do not show an increase of binding energy [START_REF] Audi | The nubase2012 evaluation of nuclear properties[END_REF]. However, 68 Ni mimics the characteristics of a doubly magic nucleus with a first 2 + state at 2 MeV excitation energy, a minimum of the B(E2, 0 + →2 + ) and a first excited state being a 0 + state. Following the scheme of shape coexistence driven by a π p-h excitation, deformed states would consist in promoting a proton from the f 7/2 to the p 3/2 , f 5/2 or p 1/2 orbitals coupled to a high -j neutron intruder orbits (in the present case the g 9/2 ). The later should coexist with spherical configuration and even become the ground state when proton are continuously removed from the πf 7/2 orbit. In 67 Co, a β-decay isomer was discovered and interpreted as a π(1p-2h) excitation across Z=28 and shape isomer [START_REF] Pauwels | Shape isomerism at n = 40: Discovery of a proton intruder state in 67 Co[END_REF]. Using the energy of the intruder π(2p-1h) state in 69 Cu and, symmetrically, the π(1p-2h)state in 67 Co, which both lie at N=40, D. Pauwels et al. have been able to predict a deformed π(2p-2h) state in 68 Ni. The predicted energy from this analysis leads to a rather low value of 2.2 MeV excitation energy, which can be understood only with an important gain in binding energy from the π -ν residual interactions between the two proton holes and the active valence neutrons across N=40 [START_REF] Pauwels | Pairing-excitation versus intruder states in 68 Ni and 90 Zr[END_REF]. The shape coexistence in 68 Ni has been investigated by the observation of low lying 0 + state as a candidate for the deformed proton configuration.

The systematics of the 2 + excitation energy of Fe isotopes doesn't show any local maximum at N=40 but a continuous decrease from N=36 to N=58, as an indication of collectivity and therefore deformation. In comparison with N=28, this onset of collectivity occurs closer to the proton shell closure (see figure 3.1). As Ar and S isotopes show a local maximum at N=28, already 2 protons from 68 Ni, no increase of the excitation energy is observed. In Co isotopes, similarly to Cu and Zn in which coexist single particle, collective and core-coupled states [START_REF] Stefanescu | Interplay between single-particle and collective effects in the odd-a cu isotopes beyond n = 40[END_REF][START_REF] C˘eliković | Probing collectivity in zn isotopes with one particle or hole outside the n = 40 subshell closure[END_REF], are states that would be naturally expected in odd-A masses based on the coupling of the single proton hole in the f 7/2 with even-even cores states. The corresponding multiplet is composed of the f 7/2 ⊗ 2 + = (3/2 -, 9/2 -, 11/2 -) states.

The systematics of excitation energies and B(E2) values in the Fe, Co and Ni isotopes show that the 9/2 - 1 (Co) and 2 + 1 (Ni) excitation energies as well as those of the 3/2 - 1 (Co) and 2 + 1 (Fe) states behave similarly. It is therefore tempting to interpret the 9/2 - 1 state as a proton hole coupled to 2 + 1 (Ni) [START_REF] Pauwels | Structure of 65,67 Co studied through the β decay of 65,67 Fe and a deep-inelastic reaction[END_REF] and the 3/2 - 1 state as a proton particle coupled to the deformed 2 + 1 (Fe). The 11/2 - states have been recently established and not shown in the figure. Weakly deformed excited states must have all the characteristics of a spherical core-coupled states (weak coupling scheme). The barycentre of the multiplet should equal the 2 + energy of the even-even Ni core and the B(E2) to the ground state. Deformed states should follow the strong coupling scheme where the intruder neutron orbit νg 9/2 is coupled to the rotational band of the even-even Fe core. The deformed configuration in odd-Co should present a (2 + ⊗ g 9/2 ) = 11/2 + , (4 + ⊗ g 9/2 ) = 13/2 + etc ... sequence similar to the 2 + , 4 + etc ... rotational band in the even-even Fe core. The low energy excited states 9/2 -, 11/2 -and 3/2 -states have been identified in 63,65,67 Co. In the present study, the core-coupled scheme was tested by the systematic lifetime measurements of relevant excited states leading to reduced transition probabilities and compared to large scale shell model calculations.

Lifetime measurement and transition probabilities around N=40

The onset of collectivity in the mass region was probed by the measurement of the reduced transition probabilities (B(E2) and B(M1)) from lifetime measurements using the plunger technique. A series of experiments were conducted at the GANIL facility using the EXOGAM and AGATA HPGe arrays, the VAMOS magnetic spectrometer and a plunger device to measure lifetime of excited states in neutron rich Fe and Co isotopes produced by multi-nucleon transfer between heavy ions. Lifetime in 62,64 Fe 2 + and 4 + states have shown that by comparison with shell model calculations, the νg 9/2 occupancy triggers the onset of collectivity at N=40 as the full amplitude is reproduced when the νd 5/2 is included [START_REF] Ljungvall | Onset of collectivity in neutron-rich fe isotopes: Toward a new island of inversion?[END_REF][START_REF] Klintefjord | Measurement of lifetimes in 62,64 Fe, 61,63 Co, and 59 Mn[END_REF]. The core coupled assumption in the (3/2 -, 9/2 -, 11/2 -) states in Co isotopes was tested by the measurement of their B(E2) to the ground state (tentatively assigned to J π =(7/2 -) from the single proton hole in the f 7/2 orbit) and compared to the E2 transition in the corresponding 2 + state in the even-even core. In the following a review of the measured values in Co is presented and enriched from measured performed in other facilities.

The 3/2 - 1 state

The B(E2; 3/2 - 1 → 7/2 - 1 ) value in 63 Co and 65 Co have been deduced from the lifetime measurement in [START_REF] Dijon | Lifetime measurements in 63 Co and 65[END_REF][START_REF] Olaizola | Properties of low-lying states in 65 Co from lifetime measurements[END_REF]. The measured value in 63 Co is found well below that of B(E2; 2 + 1 → 0 + 1 ) in Fe. This indicates that the N = 40 gap has not completely collapsed at Z = 27 or that the 3/2 - 1 state has in fact a single-particle character. As shown in [START_REF] Ljungvall | Onset of collectivity in neutron-rich fe isotopes: Toward a new island of inversion?[END_REF][START_REF] Klintefjord | Measurement of lifetimes in 62,64 Fe, 61,63 Co, and 59 Mn[END_REF], the onset of collectivity in Fe is clearly established at N = 38. If the 3/2 - 1 state results from the coupling of a proton to the 2 + 1 (Fe) state, then the B(E2; 3/2 - 1 → 7/2 - 1 ) value should rise at N = 38, making the measurement of this quantity in 65 Co very important. The 3/2 -lifetime was recently measured in 65 Co [START_REF] Olaizola | Properties of low-lying states in 65 Co from lifetime measurements[END_REF] with a value τ =5(5) ps leading also to a B(E2) = 17(16) W.u to be compared to 22(2) W.u for the B(E2, 2 + → 0 + ) in 64 Fe and 7(1) W.u in 66 Ni making the experimental conclusion elusive. It is however presently proposed that it belongs to the πf -1 7/2 ⊗ 2 + Ni state multiplet [START_REF] Olaizola | Properties of low-lying states in 65 Co from lifetime measurements[END_REF].

The 9/2 - 1 state

The lifetime to the ground state was measured in 63,65 Co in [START_REF] Dijon | Lifetime measurements in 63 Co and 65[END_REF]. The 9/2 - 1 → 7/2 - 1 transition probability is even more difficult to interpret since the decay occurs by E2 or M1, and the interpretation depends on the assumed multi-polarity. Presently the E2/M1 mixing ratio is not known. Assuming a pure E2 transition, the B(E2; 9/2 - 1 → 7/2 - 1 ) value extracted for 63 Co is compatible with the B(E2; 2 + 1 → 0 + 1 ) value in 64 Ni. It is proposed that it belongs to the πf -1 7/2 ⊗ 2 + Ni state multiplet in [START_REF] Dijon | Lifetime measurements in 63 Co and 65[END_REF][START_REF] Olaizola | Properties of low-lying states in 65 Co from lifetime measurements[END_REF].

The 11/2 - 1 state

In [START_REF] Modamio | Lifetime measurements in neutron-rich 63,65 co isotopes using the agata demonstrator[END_REF][START_REF] Klintefjord | Measurement of lifetimes in 62,64 Fe, 61,63 Co, and 59 Mn[END_REF][START_REF] Olaizola | Properties of low-lying states in 65 Co from lifetime measurements[END_REF], the lifetime of 11/2 -states at 1664 (1674) (1480) keV in 61 Co ( 63 Co) ( 65 Co) respectively have been measured and were interpreted also as a πf -1 7/2 ⊗ 2 + Ni state based on the deduced transition strength similar to the B(E2,2 + → 0 + ) in Ni.

The 1/2 - 1 state

This state plays a major role in the shape coexistence scenario. In [START_REF] Pauwels | Shape isomerism at n = 40: Discovery of a proton intruder state in 67 Co[END_REF], D. Pauwels et al. reported the observation of a J π = (1/2 -) β isomeric state at 491 keV in 67 Co and interpreted as a deformed proton intruder state 1p-2h across the Z=28 shell gap. A (1/2 -) proton state is proposed at 1095 keV in 65 Co [START_REF] Pauwels | Structure of 65,67 Co studied through the β decay of 65,67 Fe and a deep-inelastic reaction[END_REF]. The E2/M1 mixing ratio of the transition to the 3/2 -state is unknown [START_REF] Olaizola | Properties of low-lying states in 65 Co from lifetime measurements[END_REF].

Assuming a pure E2 transition to the 3/2 - 1 state, the deduced B(E2) from its lifetime is found large, B(E2) = 67(1) W.u. On the contrary, a pure M1 multi-polarity leads to low B(M1). It is presently interpreted as a member of the deformed configuration [START_REF] Olaizola | Properties of low-lying states in 65 Co from lifetime measurements[END_REF]. This deformed state is not populated in multi-nucleon transfer reaction [START_REF] Dijon | Lifetime measurements in 63 Co and 65[END_REF][START_REF] Recchia | Configuration mixing and relative transition rates between low-spin states in 68 ni[END_REF].

Summary of the experimental results

The measured lifetimes are summarized in table 3.1 and deduced transition probabilities in table 3.2. The experiments in [START_REF] Dijon | Lifetime measurements in 63 Co and 65[END_REF][START_REF] Klintefjord | Measurement of lifetimes in 62,64 Fe, 61,63 Co, and 59 Mn[END_REF][START_REF] Modamio | Lifetime measurements in neutron-rich 63,65 co isotopes using the agata demonstrator[END_REF] correspond to plunger measurements in multi-nucleon transfer reaction using different heavy beams and targets. Even if the range in sensitivity is similar, they differ in populated states making them complementary. The reference [START_REF] Olaizola | Properties of low-lying states in 65 Co from lifetime measurements[END_REF] is a fast-timing measurement after β-decay populating different states. From these set of measurements, it is concluded that the (3/2 -), (9/2 -) and (11/2 -) states are part of the πf -1 7/2 ⊗ 2 + Ni multiplet based on the similarities in excitation energy and corresponding B(E2) to the ground state. This conclusion is supported by comparison with shell model calculations. However, one could argue that the experimental conclusion is overstated regarding the uncertainties highlighted in table 3.2. First, the B(E2) for the (9/2 -) state in 63 Co is not determined accurately since the δ(E2/M 1) mixing ratio is not measured. The (3/2 -) in 65 Co has 100% error bar and the relatively precise value for the (3/2 -) and (11/2 -) states in 63 Co differ significantly from the B(E2) in 64 Ni. Finally, only the (11/2 -) state in 65 Co fits well the weak coupling scheme. More precise measurements are there-fore needed to probe experimentally the coupling scheme and track possible deviation from the model to probe the influence of the deformed configurations. Spins and parities must be uniquely determined and δ(E2/M 1) mixing ratio measured. Finally, higher precision lifetime measurements or Coulomb excitation cross-sections are mandatory to extract with smaller error bar the reduced transition probabilities.

The (1/2 -) is interpreted in comparison with shell model calculation as a deformed proton state involving excitation above the Z=28 gap in [START_REF] Olaizola | Properties of low-lying states in 65 Co from lifetime measurements[END_REF]. There is, today, no unambiguous experimental data supporting the interpretation. Indeed, the long lifetime measured in [START_REF] Olaizola | Properties of low-lying states in 65 Co from lifetime measurements[END_REF] arises mainly from the low energy E2 transition to the (3/2 -) state and cannot be used as an argument on its structure since the δ(E2/M 1) mixing ratio is not measured. A (1/2 -) is not part of the πf -1 7/2 ⊗ 2 + Ni multiplet and in the interpretation of a large deformation, the (1/2 -) spin parity can be related to a single proton excitation to the p 1 3/2 (1/2 -[321])orbit. One could speculate that when neutron are added to the g 9/2 orbit, this state could become the ground state as a result of the onset of deformation established in the Fe isotopes. The ground state spin and parity beyond N=40 in Co isotopes remain unknown. The possible spin/parity inversion of the ground state from 7/2 -→ 1/2 -, as suggested by the excitation energy systematic, is not yet established.

Theoretical calculations

Calculations from [START_REF] Dijon | Lifetime measurements in 63 Co and 65[END_REF] are compared with the data in [START_REF] Dijon | Lifetime measurements in 63 Co and 65[END_REF]. It is clear that the inclusion of the 1g 9/2 orbital becomes imperative as the neutron number approaches N = 40. This conclusion can be drawn by observing that, towards N = 40, excitation energies are overestimated in a pf model space and B(E2) values are underestimated. Nevertheless, even with the pfg interaction which includes the 1g 9/2 orbital, the excitation energies of the 3/2 - 1 and 9/2 - 1 levels are slightly higher and the B(E2) values of the transitions to the ground state are somewhat smaller in 67 Co compared to lighter Co isotopes, indicative of some effect of an N = 40 sub-shell closure in the calculation. Our measured B(E2; 3/2 - 1 → 7/2 - 1 ) value in 63 Co falls well below the value calculated with pfg, which indicates that the shell model overestimates the collectivity of the 3/2 - 1 state. On the other hand, the calculated B(E2; 9/2 - 1 → 7/2 - 1 ) value in 63 Co comes out at 11.1 W.u, in excellent agreement with the experimental value of 11.4 (55) W.u, obtained under the assumption of pure E2. However, although the E2 rate is well reproduced by the shell model, it fails to obtain the correct B(M1) value for this transition. This can be inferred from the lifetime of the 9/2 - 1 level which is calculated a factor 10 shorter than the measured value of 0.9 (4) ps due to a large calculated B(M1; 9/2 - 1 → 7/2 - 1 ) value of 0.36 W.u, obtained with bare nucleon g factors. This result is not significantly altered if the spin g factors are quenched by a factor 0.7.

To what extent are the simple interpretations, mentioned in the introduction, of a 1πf 7/2 particle coupled to a 0 + 1 or 2 + 1 in Fe or a (1πf 7/2 ) -1 hole coupled to a 0 + 1 or 2 + 1 in Ni, borne out by the shell-model calculations? This can be illustrated with the example of the 7/2 - 1 and 9/2 - 1 states.

The most important components of these states are

|7/2 - 1 ≈ 0.62|(1πf 7/2 ) -1 ⊗ 0 + 1 (Ni); 7/2 ± 0.38|(1πf 7/2 ) -1 ⊗ 2 + 1 (Ni); 7/2 + • • • , |9/2 - 1 ≈ 0.62|(1πf 7/2 ) -1 ⊗ 2 + 1 (Ni); 9/2 ± 0.31|(1πf 7/2 ) -1 ⊗ 4 + 1 (Ni); 9/2 + • • • . (3.1)
These components represent barely 50% of the total. There is thus considerable fragmentation of the wave function in the shell model and, accordingly, the simple interpretations at best are only qualitatively valid. This fragmentation also explains the large B(M1; 9/2 - 1 → 7/2 - 1 ) value obtained in the shell model. No M1 transition is allowed between the main components in the decomposition (3.1) since they involve different core states (0 + 1 and 2 + 1 , respectively). On the other hand, the first component of the 9/2 - 1 state and the second component of the 7/2 - 1 state are connected by the proton part of the M1 operator, and this leads to the large calculated B(M1) value. Given that the measured lifetime of the 9/2 - 1 level implies a small B(M1; 9/2 - 1 → 7/2 - 1 ) value, a tentative conclusion of our analysis is that the shell model using these interaction predicts too much fragmentation. No new recent measurement of the 3/2 -(Fe-like) and 9/2 -(Ni-like) lifetime have been reported since.

The measured B(E2) were finally compared to state-of-the-art Shell Model calculations using the LNPS interaction [START_REF] Lenzi | Island of inversion around cr64[END_REF]. Calculations reproduce well the experimental level scheme [START_REF] Recchia | Spectroscopy of odd-mass cobalt isotopes toward the n = 40 subshell closure and shell-model description of spherical and deformed states[END_REF] and the measured lifetime of the 11/2 -states [START_REF] Klintefjord | Measurement of lifetimes in 62,64 Fe, 61,63 Co, and 59 Mn[END_REF]. In [START_REF] Olaizola | Properties of low-lying states in 65 Co from lifetime measurements[END_REF], the B(E2) are compared to measured values in 65 Co and in particular to the proton deformed (1/2 -) state. Experimentally, its B(E2) to the 3/2 -is surprisingly high, assuming a pure E2 transition. This large B(E2) is not compatible with the calculated value. This is unexpected since the interaction has reproduced well the spectroscopic data in the mass region. However, it must be underlined that the experiment was not design to measure the M1/E2 mixing ratio. Assuming instead a pure M1 transition, the calculated B(M1) is 6.10 -3 Wu to be compared to 2.10 -2 Wu in LNPS which is reasonably close. This is supporting that the transition is dominated by the M1 component and the corresponding B(E2) probably low. To support the dominant M1 contribution, the Weisskopf estimates can be used. The Weisskopf estimates for the 212.7 keV decay to the (3/2 -) state in 65 Co are τ = 123 ns for an E2 and τ = 3 ps for an M1. The measured value is τ = 1.80(2) ns. Using the E2 Weisskopf estimates, a mixing ratio δ(E2/M1) ∼ 0.01 is calculated and is supporting a large domination of the M1 component. The (1/2 -) state is interpreted as a deformed proton intruder state. In the even-even Ni core, such intruder appears as a (2p-2h) 0 + state [START_REF] Pauwels | Shape isomerism at n = 40: Discovery of a proton intruder state in 67 Co[END_REF]. The shape isomers will be discussed in the next section but as the collectivity of excited states in Co is discussed, a comparison between the 0 + def. → 2 + 1 in the even-even Ni core and the (1/2 -) → (3/2 -) will be discussed in the following.

In 66 Ni, the deformed 0 + 4 proton state is located at 2974 kev with a B(E2) to the 2 + 1 state of 0.2 W.u. [START_REF] Leoni | Multifaceted quadruplet of low-lying spin-zero states in 66 Ni: Emergence of shape isomerism in light nuclei[END_REF]. In [START_REF] Olaizola | Search for shape-coexisting 0 + states in 66 Ni from lifetime measurements[END_REF], the LNPS calculations leads to a low B(E2)=8.10 -3 W.u. The MCSMC from [START_REF] Leoni | Multifaceted quadruplet of low-lying spin-zero states in 66 Ni: Emergence of shape isomerism in light nuclei[END_REF] predicts a B(E2)= 6.10 -3 W.u. Both experimental and calculations are coherent with a low B(E2). The (1/2 -) → (3/2 -) B(E2) in 65 Co is also found small. In 68 Ni, the deformed proton state is located at 2511 keV (see next section). According to [START_REF] Crider | Shape coexistence from lifetime and branching-ratio measurements in 68,70ni[END_REF], the B(E2) to the 2 + 1 is 2.3 W.u. The published LNPS calculations are 0.9 W.u [START_REF] Crider | Shape coexistence from lifetime and branching-ratio measurements in 68,70ni[END_REF][START_REF] Dijon | Discovery of a new isomeric state in 68 ni: Evidence for a highly deformed proton intruder state[END_REF] in reasonable agreement. The LNPS wave function for the 1/2 -state at 1290 keV in 65 Co has been published in [START_REF] Olaizola | Properties of low-lying states in 65 Co from lifetime measurements[END_REF] and has great similarities with the wave function of the 0 + proton deformed states in 66 state in Co corresponds to a 1 proton excitation from the f 7/2 to the f 5/2 and neutrons excitations from f p to g 9/2 .

Shape Isomers and deformed state in the vicinity of 68 Ni

The quest for a π(2p-2h) 0 + intruder state in 68 Ni is a long story. The first speculation for a shape isomer in this nuclei was published in [START_REF] Bernas | Magic features of 68ni[END_REF][START_REF] Girod | Spectroscopy of neutron-rich nickel isotopes: Experimental results and microscopic interpretation[END_REF]. Using the 70 Zn( 14 C, 16 O) 68 Ni reaction, excited states have been populated in a pure 2 protons transfer. Two excited states at 1.77 MeV and 2.20 MeV have been identified and interpreted by differential cross section to be the 0 + 2 and 2 + 1 states respectively. The 2 + 1 was later proposed at 2033 keV. The presence of a 0 + state as first excited state triggered discussion on its nature and the mean-field calculations using the Gogny interaction display a spherical minimum, but a second minimum exists at large deformation β = 0.3. This minimum could correspond to the observed 0 + 2 isomeric state. The excitation energy of the 0 + 2 isomeric state has been recently remeasured and placed at an energy of 1604 keV (instead of 1770 keV) based on the direct detection of the 0 + 2 → 0 + 1 E0 transition [START_REF] Suchyta | Shape coexistence in 68 Ni[END_REF] and was interpreted as a neutron state with 2 particles scattered from the p 1/2 into the g 9/2 of moderate oblate deformation (β ∼0.2) [START_REF] Suchyta | Shape coexistence in 68 Ni[END_REF][START_REF] Recchia | Configuration mixing and relative transition rates between low-spin states in 68 ni[END_REF]. The 66 Ni(t,p) 68 Ni reaction was performed recently and 2 neutron transfer cross sections have shown a strong population of the 0 + 1 state and a weak population of the 0 + 2 state [START_REF] Elseviers | Probing the semi-magicity of 68ni via the 66ni(t,p)68ni two-neutron transfer reaction in inverse kinematics[END_REF][START_REF] Flavigny | Microscopic structure of coexisting 0 + states in 68 Ni probed via two-neutron transfer[END_REF]. This result is consistent with the νg 2 9/2 interpretation of the 0 + 2 state.

Despite the large number of experiments dedicated to 68 Ni, no evidence for a 0 + state based on a proton-pair excitation could be found. In [START_REF] Mueller | β decay of 66 Co, 68 Co, and 70[END_REF], W. F. Mueller et al. proposed a (0 + ) state at 2511 keV with a unique γ decay to the 2 + 1 state, populated in the decay of a low spin (3 + ) β isomer (T 1/2 =1.6(3) s.) in 68 Co. The state in 68 Ni was not populated in the β-decay of the (7 -) ground state. The proposed spin/parity for the (3 + ) isomer leads to an interpretation of a neutron state based on a pair excitation from the f 5/2 into the g 9/2 for the 2511 keV state. In [START_REF] Pauwels | Shape isomerism at n = 40: Discovery of a proton intruder state in 67 Co[END_REF], D. Pauwels et al. reported the observation of a J π = (1/2 -) β isomeric state at 491 keV in 67 Co and interpreted as a deformed proton intruder state (1p-2h) across the Z=28 shell gap. The proposed wave function for the proposed (1/2 -) state is a π(f -2 7/2 p +1 3/2 ) deformed configuration with the Nilsson 1/2 -[321] orbital. The coupling with neutron excited from the pf to the intruder gd shell give rise to the reduction of the Z=28 effective shell gap and to the low excitation energy of the intruder state. This isomeric state associated to the 7/2 -π(2p-1h) in 69 Cu at 1711 keV allows the prediction of a π(2p-2h) deformed state at 491+1711 = 2202 keV in 68 Ni [START_REF] Pauwels | Pairing-excitation versus intruder states in 68 Ni and 90 Zr[END_REF]. In [START_REF] Dijon | Discovery of a new isomeric state in 68 ni: Evidence for a highly deformed proton intruder state[END_REF], we reported the observation of delayed 168(1) keV γ-ray transition with T 1/2 = 216 +65 -50 ns in 68 Ni populated by the two proton transfer reaction 238 U( 70 Zn, 68 Ni) 240 Pu. We proposed as assignment for this E γ =168(1) keV transition the decay from a new 0 + 3 state at 2202(1) keV to the 2 + 1 . Because of the very limited statistics, it was unfortunately not possible to use γ-γ coincidences to build the level scheme. The non-observation of the 2 + 1 → 0 + 1 decay is consistent both with the very low cross-section to populate this new isomer and with the low efficiency at the focal plane at 2 MeV. Weisskopf estimates for the half-life of an E1, E2, M1 and M2 168 keV transition are 0.1 ps, 300 ns, 6 ps and 20 µs respectively. The measured half-life indicates therefore an E2 character. The corresponding very low B(E2 : 0 intruder state deduced by D. Pauwels et al. [START_REF] Pauwels | Pairing-excitation versus intruder states in 68 Ni and 90 Zr[END_REF]. In [START_REF] Dijon | Discovery of a new isomeric state in 68 ni: Evidence for a highly deformed proton intruder state[END_REF], we performed large shell model (LSSM) calculations with the most recent interaction tailored for the mass region (LNPS) and in an extended valence space for 68 Ni [START_REF] Lenzi | Island of inversion around cr64[END_REF]. In these calculations, a third 0 + 3 state is found at 2.4 MeV for which the dominant configuration is a pure π(2p-2h) i.e. (f 7/2 ) -2 . This intruder state has only pure normal parity neutron states as neighbours. The 2 + 1 state wave function is calculated to be a pure ν(2p-2h) configuration and hence with a negligible overlap with the proposed state. Therefore the B(E2) is expected to be very small, possibly leading to an isomeric character as observed. The calculations show that the strong deformation in the Cr chain is the result of the proton-neutron correlations between the neutrons populating the 0g 9/2 -1d 5/2 orbitals and the valence protons in the pf shell. The calculated deformation for the 0 + 1,2,3 states are β 2 ≃ 0.18, 0.24 and 0.4 respectively. The shell model calculations using LNPS published in [START_REF] Dijon | Discovery of a new isomeric state in 68 ni: Evidence for a highly deformed proton intruder state[END_REF] give for the first time the microscopical description of the deformed intruder proton state. The 0 + 2 is a pure neutron state with 2 neutron in the g 9/2 from the pf shells. The 0 + 3 is a strict excitation of proton pair from the f 7/2 into the f 5/2 with a fragmented neutron wave function in the pf -g 9/2 orbitals as a signature of large deformation. In the calculations, a fourth 0 + state compatible with the state observed by W. F. Mueller et al. proposed at 2511 keV [START_REF] Mueller | β decay of 66 Co, 68 Co, and 70[END_REF] is not predicted.

ν (g ) 9/2 7/2 -2 (f ) π 0 + ν (g ) 9/2 7/2 -2 (f ) π + 0 2 + ν (g ) 9/2 7/2 -2 (f ) π ν (g ) 9/2 7/2 -2 (f ) π (g )
Following the publication, several experiments focussed on the spectroscopy of 68 Ni. The new isomer was not confirmed by [START_REF] Chiara | Low-spin states and the non-observation of a proposed 2202-kev, 0 + isomer in 68 ni[END_REF] and a delayed 168 keV γ-ray in coincidence with the 2 + 1 decay was excluded. However, the spin parity assignment of the (0 + ) state at 2511 keV proposed by [START_REF] Mueller | β decay of 66 Co, 68 Co, and 70[END_REF] was confirmed by the 0 → 2 → 0 angular correlation measurement. Presently, 2 excited 0 + states are confirmed in 68 Ni. The conclusion of [START_REF] Chiara | Low-spin states and the non-observation of a proposed 2202-kev, 0 + isomer in 68 ni[END_REF] is "Particular attention was devoted to searching for the decay of the proton-2p2h, 2202-keV, 0 + isomer. No evidence for such an intruder state was observed in the present study. Given the impact such a state would have on understanding the structure of nuclei in the 68 Ni region, more conclusive evidence of its location is required." . The 2511 keV 0 + 3 state is compatible with the shell model prediction for a 2p-2h proton intruder state and the results of the other experiments, even if its shape and microscopical configuration was never measured, support the shape isomer interpretation [START_REF] Dijon | Discovery of a new isomeric state in 68 ni: Evidence for a highly deformed proton intruder state[END_REF][START_REF] Recchia | Configuration mixing and relative transition rates between low-spin states in 68 ni[END_REF][START_REF] Suchyta | Shape coexistence in 68 Ni[END_REF][START_REF] Flavigny | Characterization of the low-lying 0 + and 2 + states in 68 Ni via β decay of the low-spin 68 Co isomer[END_REF][START_REF] Crider | Shape coexistence from lifetime and branching-ratio measurements in 68,70ni[END_REF]. It is worth mentioning that the E0 branch from the 0 + 3 decay was not observed [START_REF] Flavigny | Characterization of the low-lying 0 + and 2 + states in 68 Ni via β decay of the low-spin 68 Co isomer[END_REF][START_REF] Crider | Shape coexistence from lifetime and branching-ratio measurements in 68,70ni[END_REF] but its lifetime was established at T 1/2 =0.57(5) ns. Using upper limit for the E0 branching ratios to the 0 + 2 and 0 + 1 states, small ρ 2 (E0) ≤ 5 m.u and ≤ 25 m.u. respectively were extracted. The low ρ 2 (E0) could be interpreted as a signature of small differences in deformation between state or more likely as a signature of a low mixing of the respective wave functions. In [START_REF] Recchia | Configuration mixing and relative transition rates between low-spin states in 68 ni[END_REF][START_REF] Flavigny | Characterization of the low-lying 0 + and 2 + states in 68 Ni via β decay of the low-spin 68 Co isomer[END_REF], a 2 + 2 state at 2742.6 keV is proposed as band member of the 0 + 3 intruder state. Such rotational band with a E γ (2 + → 0 + ) = 232 keV in 68 Ni leads to an approximated deformation parameter β 2 ∼ 0.45 using the Ramans estimate, compatible with the LNPS calculations.

In the present understanding, the 68 Ni ground state configuration has a moderate deformation. The 2 + 1 and 0 + 2 states correspond to a neutron pair scattered into the g 9/2 orbital. The configuration of the highly deformed intruder 0 + 3 state, with valence protons holes in the pf shell, is very similar to the Fe (0 + 1 /2 + 1 ) states, i.e. with two proton holes in the πf 7/2 orbital. Therefore, the configuration corresponding to the 0 + 3 state in 68 Ni migrates down to 491 keV excitation energy in 67 Co (the (1/2 -) isomer) and becomes the ground state in 66 Fe when removing protons to the Z=28 shell closure. The normal configuration, by opposition to the intruder one, might then appear as a low-lying 0 + state in 66 Fe. In [START_REF] Olaizola | Beta decay of 66mn to the n = 40 nucleus 66fe[END_REF][START_REF] Liddick | Low-energy level schemes of 66,68 fe and inferred proton and neutron excitations across Z = 28 and N = 40[END_REF][START_REF] Olaizola | Search for shape-coexisting 0 + states in 66 Ni from lifetime measurements[END_REF], a 0 + 2 state at 1414 keV was identified in the β-decay of 66 Mn. The shell model calculations predict such a state at the proper energy with a normal configuration πf 5 7/2 (p 3/2 f 5/2 p 1/2 ) 1 νp 4 3/2 f 4 5/2 p 2 1/2 g 2 9/2 similar to the (0 + 2 /2 + 1 ) states in 68 Ni [START_REF] Olaizola | Beta decay of 66mn to the n = 40 nucleus 66fe[END_REF]. As a summary, this shape coexistence scenario is illustrated in figures 3.4 and 3.3.

Recent large scale shell model calculations have provided wave function description of the 0 + states in 68 Ni. The LNPS [START_REF] Dijon | Discovery of a new isomeric state in 68 ni: Evidence for a highly deformed proton intruder state[END_REF][START_REF] Flavigny | Characterization of the low-lying 0 + and 2 + states in 68 Ni via β decay of the low-spin 68 Co isomer[END_REF] and MCSMC [START_REF] Tsunoda | Novel shape evolution in exotic ni isotopes and configuration-dependent shell structure[END_REF], state-of-the-art, are compared in figure 3.6. Even if the approaches are slightly different, the proposed configurations are very similar for the 0 + 3 state with a pair excitation of proton from the f 7/2 into the f 5/2 and fragmented neutron wave function with significant increase of the g 9/2 occupancy. The p-n interaction between the πf -2 7/2 and νg 3 9/2 gives rise to the deformed configuration. It is worth mentioning that since this intense experimental work on 68 Ni, similar proton deformed configurations have been proposed in 66 Ni [START_REF] Leoni | Multifaceted quadruplet of low-lying spin-zero states in 66 Ni: Emergence of shape isomerism in light nuclei[END_REF][START_REF] Olaizola | Search for shape-coexisting 0 + states in 66 Ni from lifetime measurements[END_REF][START_REF] Stryjczyk | βdecay study of the 66 Mn-66 Fe-66 Co-66 Ni decay chain[END_REF] and in 70 Ni [START_REF] Chiara | Identification of deformed intruder states in semimagic 70 Ni[END_REF][START_REF] Prokop | New low-energy 0 + state and shape coexistence in 70 Ni[END_REF][START_REF] Morales | Type ii shell evolution in a=70 isobars from the n 226540 island of inversion[END_REF] giving rise to a triple shape coexistence scenario evolution between 66 Ni and 70 Ni. The question of low lying deformed 0 + state in the neutron mid-shell in 72,74 Ni is therefore opened. From [START_REF] Leoni | Multifaceted quadruplet of low-lying spin-zero states in 66 Ni: Emergence of shape isomerism in light nuclei[END_REF], four 0 + states have been identified in 66 Ni. How these 3 excited 0 + states evolve in 68 Ni leaves room for a possible explanation to the 168 keV delayed γ-ray transition.

Even if the experimental data show consistency in even-even nuclei, the β-decay chain for the mass A=68 deserves some attention since the spin parity of the β-isomer in 68 Co plays a major role in the possible constraint of the microscopic configuration of the proton 2p-2h intruder state in 68 Ni. After the discovery of the short lived β-isomer in 68 Co [START_REF] Mueller | β decay of 66 Co, 68 Co, and 70[END_REF], several spin/parity assignments were proposed in the light of the populated excited states in 68 Ni. The claim for a proton 0 + 2p-2h state at 2511 keV in 68 Ni should make a coherent β-decay chain for A=68. This review is proposed below. The list of the different assumptions for the spins and parities assignement is chronological.

• (7 -) high spin ground state β-decay. In [START_REF] Mueller | β decay of 66 Co, 68 Co, and 70[END_REF], the high spin ground state T 1/2 =0.23(3) s. is proposed as a 7 -state with the πf -1 7/2 ⊗ νg +1 9/2 configuration. The decay of this state doesn't populate the 0 + 3 state in 68 N i but only pure neutron states. Indeed, the νf 5/2 → πf 7/2 Gamow-Teller is favoured and populates closed proton shell in 68 Ni with neutron states having the νf -1 5/2 g +1 9/2 configuration compatible with the (6 -, 7 -) states assumptions. • (3 + ) low spin isomer assumption. In [START_REF] Mueller | β decay of 66 Co, 68 Co, and 70[END_REF], the low spin isomer T 1/2 =1.6(3)s. state is proposed Figure 3.6: Proton and neutron occupancy for the 0 + states in 68 Ni for (left) LNPS from [START_REF] Dijon | Discovery of a new isomeric state in 68 ni: Evidence for a highly deformed proton intruder state[END_REF][START_REF] Flavigny | Characterization of the low-lying 0 + and 2 + states in 68 Ni via β decay of the low-spin 68 Co isomer[END_REF] and (right) MCSMC from [START_REF] Tsunoda | Novel shape evolution in exotic ni isotopes and configuration-dependent shell structure[END_REF] as a (3 + ) state with the πf -1 7/2 ⊗ νg +2 9/2 p -1 1/2 configuration. The decay of this state shows a weak population of the 0 + 3 state in 68 N i. Similarly to the (7 -) ground state, a (3 + ) would favour a νf 5/2 → πf 7/2 Gamow-Teller, populating pure neutron states having the νp -1

1/2 f -1 5/2 g +2 9/2
configuration. According to shell model calculations, this is the exact configuration of the 0 + 2 and 2 + 1 states in 68 Ni. The decay of the 0 + 2 was not observed in [START_REF] Mueller | β decay of 66 Co, 68 Co, and 70[END_REF] but the branching ratio to the 2 + 1 is the largest (≤39(8)%). Proton states at 5 MeV in 68 Ni are also populated indicating that there is a significant νp 1/2 → πp 3/2 Gamow-Teller strength leading to proton states with the πf -1 7/2 p +1 3/2 ⊗ νp -2 1/2 g +2 9/2 configuration. Without the gain in pairing energy, such 1p-1h π states lies at higher excitation energy than a 2p-2h state. The weak population of the proposed 2p-2h proton state in 68 Ni would involve the combination of a f 5/2 → πf 5/2 Gamow-Teller with a f 7/2 → f 5/2 proton excitation. This could be compatible with the low branching ration to the 0 + 3 assuming the 2p-2h proton configuration.

• (1 + ) low spin isomer assumption. In [START_REF] Liddick | Low-energy structure of 66 27 co 39 and 68 27 co 41 populated through β decay[END_REF], 68 Co was populated from the 68 Fe ground state favouring the low spin isomer. It is proposed a new (1 + ) spin/parity assignment for the isomer fed by an E1 transition from a state at 45 keV above, thus a 2 -or 1 -state. The (1 + ) assignment suggests a proton excitation into the deformed intruder orbital coupled to two neutrons in the g 9/2 , ie the πf -2 7/2 p +1 3/2 ⊗ νp 1 1/2 g +2 9/2 configuration. The νp 1/2 → πp 3/2 Gamow-Teller should be favoured and populates states with the πf -2 7/2 p +2 3/2 ⊗ νp 0 1/2 g +2 9/2 configuration, ie a proton pair excitation above the Z=28 gap into the p 3/2 orbital. This configuration is not the exact configuration expected for the two proton intruder at low excitation energy since, according to calculations, the proton pair is predicted into the πf 5/2 (the gain in pairing energy is more important). But proton state should be anyhow favoured. Instead, the βfeeding shows a large feeding of the neutron states in 68 Ni. As a result, the proposed (1 + ) ground state is very unlikely. The decay by an E1 transition at 45 keV could instead lead to the population of a 1 + state that decays to a (2 -, 1 -) isomeric state, ie a reverse assignment to the one proposed in [START_REF] Liddick | Low-energy structure of 66 27 co 39 and 68 27 co 41 populated through β decay[END_REF]. This (2 -, 1 -) state assignment leads to a ν(p -2 1/2 g 3 9/2 ) ⊗ π(f -1 7/2 ) or (f -2 7/2 p +1 3/2 ) configuration. Such configuration with a fully depopulated νp 1/2 will favour the νf 5/2 → πf 7/2 Gamow-Teller and populate pure neutron states as experimentally observed.

• (2 -) low spin isomer assumption. In [START_REF] Flavigny | Characterization of the low-lying 0 + and 2 + states in 68 Ni via β decay of the low-spin 68 Co isomer[END_REF], the (2 -) spin/parity assignment is finally favoured.

But the proposed configuration is a low-lying deformed [321]1/2 -proton configuration coupled to the down-sloping [431]3/2 + and [440]1/2 + neutron orbitals originating from the spherical g 9/2 orbital, thus one of the previously proposed configuration ν(f

6 5/2 p -2 1/2 g 3 9/2 ) ⊗ π(f -2 7/2 p +1 3/2
). Such configuration will favour the νf 5/2 → πf 7/2 Gamow-Teller transition. The populated states will be ν(f 5 5/2 p -2 1/2 g 3 9/2 ) ⊗π(f -1 7/2 p +1 3/2 ). It means proton single particle state as observed in [START_REF] Mueller | β decay of 66 Co, 68 Co, and 70[END_REF] and to some extend if the excited proton in the p 3/2 decays to the f 7/2 , the 0 + 2 and 2 + 1 neutron states in 68 N i as experimentally observed. This excludes a strong population of the proton 2p-2h state by β-decay into the 68 Ni.

As a summary, neither the direct (t,p) reaction [START_REF] Flavigny | Microscopic structure of coexisting 0 + states in 68 Ni probed via two-neutron transfer[END_REF] or β-decay from the low or high spin states in 68 Co can strongly populate the 2p-2h deformed proton intruder state in 68 Ni. From this compilation, the (1 + ) assignment for the low spin isomer is very unlikely. The (3 + ) and (2 + ) assumptions are compatible with the current interpretation of the 0 + 3 state in 68 Ni. Figure 3.7 illustrates the compilation of the data used in this review. If possible, the ( 3 He, n) reaction from the deformed 66 Fe or α-transfer from 64 Fe would favour the population of the deformed state in The coherence with the 70 Zn( 14 C, 16 O) 68 N i data remains puzzling. In the two experiments by [START_REF] Girod | Spectroscopy of neutron-rich nickel isotopes: Experimental results and microscopic interpretation[END_REF] and [START_REF] Bernas | Magic features of 68ni[END_REF], the 0 + 2 and 2 + 1 states were proposed first at 1770 and 2200 keV respectively. Later measurements, as mentioned before, corrected the respective values at 1604 and 2033 keV. The two corrections are 166 and 167 keV respectively (see figure 3.8), very close to the proposed delayed line at 168 keV which was never confirmed later. This systematic shift of 167 [START_REF] Heyde | Shape coexistence in atomic nuclei[END_REF] keV is disturbing and gives room for a possible explanation for the 168 keV γ-ray observed in [START_REF] Dijon | Discovery of a new isomeric state in 68 ni: Evidence for a highly deformed proton intruder state[END_REF].

0 + 7/2 3/2 1/2 5/2 (f )(p )(f )(p ) π 9/2 3/2 5/2 1/2 (p )(f )(p )(g ) ν 7/2 3/2 1/2 5/2 (f )(p )(f )(p ) π 9/2 3/2 5/2 1/2 (p )(f )(p )(g ) ν 7/2 3/2 1/2 5/2 (f )(p )(f )(p ) π 9/2 3/2 5/2 1/2 (p )(f )(p )(g ) ν 7/2 3/2 1/2 5/2 (f )(p )(f )(p ) π 9/2 3/2 5/2 1/2 (p )(f )(p )(g ) ν
The microscopical structure of 70 Zn would reveal the possibility to populate the proper state in a strict 2 proton transfer. It is proposed to discuss, from the constrained theoretical wave functions of 70 Zn, the probability to populate the deformed proton state in 68 Ni. For this purpose, we use, as constrain, the B(E2, 2 + → 0 + ) measured using different experimental methods. The compilation of the measured B(E2,2 + 1 → 0 + 1 ) in 70 Zn is presented in figure 3.9 and compared to Shell Model calculations using the LNPS and JUN45 interactions [START_REF] Celikovic | Lifetime measurements of zn isotopes around n = 40[END_REF][START_REF] C˘eliković | Probing collectivity in zn isotopes with one particle or hole outside the n = 40 subshell closure[END_REF][START_REF] Celikovic | Nuclear structure around doubly-magic nuclei: lifetime measurements in the vicinity of 68 Ni and search for isomers around 100 Sn[END_REF]. The contributions to the ground state configurations using the JUN45 interaction are listed in table 3.3. The JUN45 calculations for the B(E2) is shown in figure 3.9 and reproduces well the experimental average value with a close Z=28 proton shell, indicating that the wave function doesn't involve any excitation from the 56 Ni core. The two main contributions are : π(p 2 3/2 f 0 5/2 )⊗ν(f 4 5/2 p 2 1/2 g 2 9/2 ) and π(p 2 3/2 f 0 5/2 )⊗ ν(f 6 5/2 p 2 1/2 g 0 9/2 ). The feeding of the proton deformed state in 68 Ni in a 2 protons stripping reaction is excluded in the second configuration since the neutron g 9/2 orbital is not occupied.

The configurations with the highest amplitude involve several neutrons scattered into the g 9/2 orbital and two protons in the p 3/2 orbital. The amplitude corresponding to an occupied f 5/2 proton orbital (**) account for 7% of the wave function. Other theoretical calculations [START_REF] Chaudhary | Study of electromagnetic properties and structure of yrast bands in neutron-rich 70 201376zn isotopes[END_REF] propose a wave function for the 70 Zn ground state as π(f 7. 4 7/2 p 1.2 3/2 f 0.7 5/2 ) ⊗ ν(f 3.7 5/2 p 0.7 1/2 g 3.3 9/2 ), ie with scattered Table 3.3: Calculated wave function for the ground state of 70 Zn using the JUN45 interaction. The configurations indicated with (**) are compatible with the population of the deformed proton state in 68 Ni by a 2 proton stripping from 70 Zn.

Amplitude 70 Zn 0.1456 ν(p 4 3/2 f 4 5/2 p 2 1/2 g 2 9/2 )π(p 2 3/2 f 0 5/2 ) 0.1275 ν(p 4 3/2 f 6 5/2 p 2 1/2 g 0 9/2 )π(p 2 3/2 f 0 5/2 ) 0.0941 ν(p 4 3/2 f 6 5/2 p 0 1/2 g 2 9/2 )π(p 2 3/2 f 0 5/2 ) 0.0677 ν(p 2 3/2 f 6 5/2 p 2 1/2 g 2 9/2 )π(p 2 3/2 f 0 5/2 ) 0.0564 ν(p
Figure 3.9: Systematics of the measured B(E2,2 + 1 → 0 + 1 ) in 70 Zn. Experimental data (a) [START_REF] Pritychenko | An update of the b(e2) evaluation for 01+ 219221+ transitions in even 2013even nuclei near n 223cz 223c28[END_REF], (b) [START_REF] Louchart | Collective nature of low-lying excitations in 70,72,74 zn from lifetime measurements using the agata spectrometer demonstrator[END_REF], (c) [START_REF] Celikovic | Lifetime measurements of zn isotopes around n = 40[END_REF], (d) [137], (e) SM JUN45, (f) SM LNPS. The average value is 299(11) e 2 f m 4 to be compared to 302 and 327 for the JUN45 and LNPS calculations respectively. neutron in the g 9/2 and also with a very low occupancy of the proton f 5/2 . The 14 C beam has a p -2 1/2 proton configuration considering a 16 O core. Therefore, the pair transfer should favour ∆l=0, and the removal of the two protons in 70 Zn from the p 2 3/2 would be favoured. The two protons stripping from the occupied p 3/2 or f 5/2 proton orbital will give rise to the wave function of the 0 + 1 and (0 + 2 and 2 + 1 ) neutron states in 68 Ni according to the proposed calculations. Therefore, the main feeding in 68 Ni should populate neutron states. The remaining four configurations with two protons in the f 5/2 orbital could populate the deformed proton state if protons are removed from the f 7/2 core with a ∆l=2 transfer, ie giving a similar 2 hole proton configuration into the f 7/2 . This transfer would correspond to a very low cross section. As a conclusion, based on the present discussion on the 70 Zn ground state wave function, it is very unlikely that the experiments performed in [START_REF] Girod | Spectroscopy of neutron-rich nickel isotopes: Experimental results and microscopic interpretation[END_REF] and [START_REF] Bernas | Magic features of 68ni[END_REF] have populated the proton deformed state in 68 Ni. In these experiments, the ground state is populated with the highest cross section, indicating that the second calculated wave function, with empty neutron g 9/2 orbital, should be the dominant configuration in 70 Zn. The 0 + 2 and 2 + 1 states are populated with two orders of magnitude less giving a very weak chance to observe the low cross section to the proton deformed state.

One could propose two reactions compatible with the population of the proton state in 68 Ni. Obviously, starting from a deformed configuration will increase the cross section. Both 64 Fe and 66 Fe ground state have a proton (f 5/2 ) -2 configuration and the lifetime measurements show that both neutron g 9/2 and d 5/2 orbitals are already populated in 64 Fe [START_REF] Ljungvall | Onset of collectivity in neutron-rich fe isotopes: Toward a new island of inversion?[END_REF]. Therefore the 64 Fe( 7 Li,t) 68 Ni (α transfer) or 66 Fe( 3 He,n) 68 Ni reactions could favor the population of the proton deformed state.

Q s systematics

Contrary to the two previous mass regions, experimental spectroscopic quadrupole moments have not been measured in the vicinity of 68 Ni (last measured is 64 Ni), in Co (last measured is 60 Co), Fe (last measured is 61 Fe) and Mn (last measured is 56 Mn) isotopes. Therefore, the large overview of spectroscopic quadrupole moments presented in the N=Z and A=100 mass region cannot be produced. This underlines the lack of experimental data in the mass region. Ni, Co, Fe and Mn elements have chemical properties making these elements difficult to extract from an ISOL target. Nuclei lying below 68 Ni are not populated by fission of heavy elements and being neutron rich are not populated in fusion-evaporation of heavy ions. These isotopes are populated by high energy heavy ions fragmentation or Multi-nucleon transfer reaction around the Coulomb barrier using heavy ions. Such area remains a Terra incognita for such techniques.

Conclusions

A shape coexistence scenario emerges around N=40 in Ni, Co, Fe and Cr isotopes. An important experimental effort in the last five years have clarified the level schemes and spin/parity assignments. Very few relevant transition probabilities have been measured and no experimental data have been published on the direct determination of the shape (ie spectroscopic quadrupole moment) or possible constrains on the microscopical configuration of the intruder state. Presently, the direct proof for a 2p-2h proton excitation would a be milestone in the mass region. In Co isotopes, unknown E2/M1 mixing prevent from a direct B(E2)/B(M1) comparison. In addition, based on the unique multi-polarity, the overlap between the B(E2) in Co and Ni is not as unambiguous as expected from the weak coupling model. The systematic in excitation energy of the (1/2 -) deformed state suggests an inversion beyond N=40 with the ground state. This has never been proven.

In 68 Ni isotope, one should remember that so far the E0 transition from the 0 + 3 was not observed and the spin assignment comes from the angular correlation 0→ 2 → 0. The shape of this state is not confirmed by a spectroscopic quadrupole moment and no B(E2) connecting to a possible rotational 2 + state is measured. The safe Coulomb excitation of a high intense 68 Ni beam would be a milestone in the present discussion.

In the last section, an analysis of the different path to reach the 2p-2h proton state by β-decay and nucleon transfer is proposed. It shows that indeed, with its proposed configuration, these direct reactions cannot strongly populate such configuration. The consistency between proton, neutron transfer and β-decay is proven. The consistency between the 2 protons transfer from 14 C and the delayed 168 keV line is not obvious. One could speculated that, instead, a third excited 0 + state, of neutron nature, similar to the one in 66 Ni, was populated.

Chapter 4

Shape coexistence around N=Z=40

Introduction

The neutron deficient isotopes, close to the N=Z line, between Ge and Zr, are a long standing area of shape coexistence. Stable Ge isotopes have shown a competition between a spherical excited configuration and a deformed ground state. As these isotopes are stable, they have been studied by safe Coulomb excitation and spectroscopic quadrupole moment for the first 2 + states were extracted. Low lying 0 +m states have been identified as evidence of shape coexistence. However, shape parameter of states built on top of this proposed shape isomers are not yet experimentally measured.

For heavier elements, from Se to Zr, Wood-Saxon calculations including a spin-orbit term [START_REF] Nazarewicz | Microscopic study of the high-spin behaviour in selected a 2243 80 nuclei[END_REF][START_REF] Gross | Band crossings and near-rigid rotation in 76kr and 78kr[END_REF] and more elaborated mean field calculations using the Skyrme interaction [START_REF] Bonche | Self-consistent study of triaxial deformations: Application to the isotopes of kr, sr, zr and mo[END_REF][START_REF] Bender | Shape coexistence in neutron-deficient kr isotopes: Constraints on the single-particle spectrum of self-consistent mean-field models from collective excitations[END_REF] or the Gogny interaction [START_REF] Libert | Microscopic descriptions of superdeformed bands with the gogny force: Configuration mixing calculations in the a=190 mass region[END_REF][START_REF] Goriely | First gogny-hartree-fock-bogoliubov nuclear mass model[END_REF][START_REF] Hilaire | Nuclear structure properties with the gogny force[END_REF][START_REF] Péru | Mean field based calculations with the gogny force: Some theoretical tools to explore the nuclear structure[END_REF][START_REF] Girod | Hartree-fock-bogoliubov predictions for shape isomerism in nonfissile even-even nuclei[END_REF][START_REF] Dechargé | Hartree-fock-bogolyubov calculations with the d1 effective interaction on spherical nuclei[END_REF] have predicted a shape coexistence between prolate and oblate configurations. Low-lying 0 +m states have been identified in 72,74,76 Se, 72,74,76 Kr and 80 Sr as first evidence. A specificity of the mass region is that several spectroscopic observations point to a strong mixing of the wave functions corresponding to the two configurations. In Kr isotopes, a perturbation of the ground state band moment of inertia at low spin was observed. Similarly, the reduced transition probabilities is showing a reduction of the collectivity at low spin as a result of the possible shape mixing (see figure 4.1). Combining the excitation energy of both the 2 + 1 and 4 + 1 states and the B(E2, 2 + 1 → 0 + 1 ), the correlation between the R 42 =4 + 1 /2 + 1 excitation energy ratio and the B(E2)[W.u.].A -1 shows that these N∼Z nuclei are located far from the systematics of all other nuclei in the nuclear chart as shown in [START_REF] Lemasson | Observation of mutually enhanced collectivity in self-conjugate[END_REF]. These off-systematics points are understood as the result of the mixing. The perturbation of the 0 + ground state and 0 +m isomeric state excitation energy with respect to the higher spins extrapolation was used to quantify for the first time the wave function mixing using the two levels mixing model [START_REF] Bouchez | New shape isomer in the self-conjugate nucleus 72 Kr[END_REF]. In the case of the neutron deficient Kr isotopes, a mixing of 27% of the wave function is extracted for the 76 Kr ground state. This mixing reaches a maximum of 50% in 74 Kr deduced from the maximum value of ρ(E0) and minimum excitation energy of the 0 +m . In 72 Kr, the 0 +m excitation energy increases with a reduced ρ 2 (E0) leading to a mixing of 10% [START_REF] Bouchez | New shape isomer in the self-conjugate nucleus 72 Kr[END_REF]. This parabolic trend suggests an inversion of shape from 78 Kr to 72 Kr with a crossing at 74 Kr where both configurations are almost degenerated [START_REF] Bouchez | New shape isomer in the self-conjugate nucleus 72 Kr[END_REF]. So far, the evaluation of the sign of the deformation, i.e. prolate or oblate, was derived from comparison with theoretical models. For instance, using a comparison between measured B(E2; 2 + → 0 + ) and theoretical values, Iwasaki et al [START_REF] Iwasaki | Evolution of collectivity in 72 Kr: Evidence for rapid shape transition[END_REF] proposed that the first 2 + state in 72 Kr is compatible with an oblate shape and therefore suggesting a prolate shape for heavier isotopes. All these experimental probes gave rise to indirect evidence of the shape coexistence. Laser spectroscopy is an experimental technique from which spectroscopic quadrupole moment in oddmass nuclei ground state can be measured as well as for long-lived isomers. Unfortunately such systematic is rather scarce close to the N=Z line and does not involve the isotopes of interest as it will be more detailed in section 4.3.

Coulomb excitation experiment results

Direct evidence of the shape coexistence in this mass region came from the safe Coulomb excitation of radioactive beams. The Coulomb excitation cross section, via the second order effect of the reorientation [START_REF] Cline | Nuclear shapes studied by coulomb excitation[END_REF]40,[START_REF] Zielińska | Analysis methods of safe coulombexcitation experiments with radioactive ion beams using the gosia code[END_REF], is sensitive to the E2 diagonal matrix element from which a spectroscopic quadrupole moment can be deduced. The very first safe multi-step coulomb excitation of a postaccelerated radioactive beam was performed for the neutron deficient Kr isotopes at the SPIRAL1 facility, GANIL in the framework of my PhD thesis work. Post-accelerated beams of 74 Kr and 76 Kr were delivered to the EXOGAM setup and bombarded a 208 Pb target just below the Coulomb barrier. A charged particle detector covered scattering angles ensuring "safe" Coulomb excitation according to Cline's criteria [START_REF] Cline | Nuclear shapes studied by coulomb excitation[END_REF]. As a result of my analysis, the first ever spectroscopic quadrupole moments in short lived excited states in unstable isotopes were measured [START_REF] Clément | Shape coexistence in neutron-deficient krypton isotopes[END_REF]. This work represents the first direct experimental proof of the proposed shape coexistence scenario in neutron deficient Krypton isotopes, and the reorientation effect has been exploited for the first time with a radioactive ion beam. The details of the analysis and discussion can be found in [START_REF] Clément | Shape coexistence in neutron-deficient krypton isotopes[END_REF]. To summarize the output of this rich set of E2 matrix elements both transitional (leading to B(E2)) and diagonal (leading to spectroscopic quadrupole moment) :

• 16(13) E2 transitional matrix elements have been measured in 76 Kr ( 74 Kr) respectively.

• The measurement has shown an inconsistent value of the 4 + 1 state lifetime in 74 Kr later confirmed and corrected by a dedicated plunger measurement [START_REF] Görgen | Lifetime measurement in 74kr and 76kr[END_REF] • 5(5) E2 diagonal matrix elements have been measured in 76 Kr ( 74 Kr) respectively for the first time.

• The transition strengths from the Coulomb excitation measurement are in good agreement with the results of the lifetime measurement.

• Using the precise lifetimes as additional input in the analysis of the Coulomb excitation data enhanced the sensitivity to the reorientation effect and allowed extracting spectroscopic quadrupole moments for several excited states.

• The negative diagonal matrix elements of the states in the ground-state bands of both isotopes prove their prolate character

• The excited 2 + 2 and 2 + 3 states are interpreted as γ-vibrational and oblate rotational states, respectively.

• However, a clear classification of the states is difficult due to the mixing of prolate and oblate configurations, on one hand, and K = 0 and K = 2 configurations, on the other hand.

• The matrix elements were interpreted in a phenomenological two-band mixing model. The results confirm the earlier finding of maximum mixing between prolate and oblate configurations in the wave functions of the 0 + states in 74 Kr. In 76 Kr, the mixing of the ground state is 31(4)% whereas it was measured at 27(1)% from the E0 spectroscopy. In 74 Kr, the mixing is found equal (52%) in both experiment.

• The purity of the wave functions increases rapidly with spin, explaining the increase in the transition strength and at the same time the very strong inter-band transitions between the different 2 + and 0 + states (see figure 4.1).

• The two level mixing model fails to describe consistently the band mixing in 76 Kr, probably due to the strong coupling with the presumed γ-vibrational band.

• The quadrupole sum rule formalism was applied to derive shape parameters for the 0 + states.

• The first rotational invariant, Q 2 (similar to |β 2 |), is found large for both states in both isotopes confirming their deformed character

• The second rotational invariant, cos3δ (similar to γ) shows that the 76,74 Kr ground state are axially prolate. The 0 + 2 state in 76 Kr is oblate with a large triaxial contribution. The second 0 + state in 74 Kr reaches a maximum of triaxility.

• The transition strengths and quadrupole moments were compared to theoretical calculations going beyond the static mean-field approach using the Gogny D1S interaction and the Skyrme SLy6 interaction. The present SLy6 interaction is limited to axial symmetry. The comparison emphasize the importance of including the triaxial degree of freedom to describe the shape coexistence in the light krypton isotopes.

Q s systematics

The diagonal E2 matrix elements extracted from the Coulomb excitation analysis can be translated into spectroscopic quadrupole moments and compared to the ground states quadrupole moment in odd-mass neighbouring nuclei. Data used for this comparison are compiled from [START_REF] Stone | Table of nuclear electric quadrupole moments[END_REF][START_REF] Campbell | Laser spectroscopy for nuclear structure physics[END_REF] and completed by quadrupole moments of the 2 + 1 states in 70,72 Se [START_REF] Hurst | Measurement of the sign of the spectroscopic quadrupole moment for the 2 + 1 state in 70 Se: No evidence for oblate shape[END_REF][START_REF] Iwasaki | Evolution of collectivity in 72 Kr: Evidence for rapid shape transition[END_REF][START_REF] Henderson | Localizing the shape transition in neutron-deficient selenium[END_REF]] and 72 Ge [START_REF] Ayangeakaa | Shape coexistence and the role of axial asymmetry in 72ge[END_REF]. In figure 4.2,the compilation of the absolute deformation amplitude (|β 2 |) is restricted to Z>31 and N<50. First, it is worth mentioning that such measurement did not yet reach the N=Z line above the Ni chain and are far from 100 Sn.

From the Z=32, a clear onset of deformation is observed between Ge and Se isotopes, i.e. when the proton f 5/2 is populated and with at least four neutrons holes in the g 9/2 . From this systematics in neutron number, the shell closure at nucleon number 40 does not present a stabilizing effect toward spherical shape and the maximum of deformation is reached at Z≃N≃38-39.

In figure 4.3, the algebraic deformation is deduced (β 2 ). The blue (green) squares indicate a prolate (oblate) deformations respectively. The first matrix shows the deformation for the ground states in odd-nucleon isotopes and for the 2 + 1 state in even-even isotopes. Beyond Se, deformation are large and prolate. Approaching the N=Z line, a prolate to oblate transition occurs in 70 Se. No other shape inversion as a function of the neutron number was proven so far. The second panel highlights the shape coexistence scenario. This matrix shows the same algebraic deformation (β 2 ) measured for isomers, which are living long-enough to allow a laser spectroscopy, and for 2 + 2,3 states built above a low-lying 0 + 2 from safe Coulomb excitation measurements. The 2 + 2 state is not chosen by default. Indeed, in this mass region, K=2 γ-band are frequent at low excitation energy. Therefore, for the compilation, the 2 The comparison between the two matrices illustrates the shape coexistence in a given isotope. In Ge isotopes, the ground state presents a very small deformation and the 2 + 3 state has a large one illustrating the shape coexistence between a spherical ground state and a deformed excited configuration. So far, the shape change in Ge isotopes was not proven. In light odd-mass Kr, the Q s in long-lived isomers are not established. Large systematics also exists for the Rb isotopes (Z=37). The ground states have large prolate deformation. The isomers have an oblate or prolate character. Beyond Rb, available data are almost non-existing.

Probing the deformation from the first excited states in even-even isotopes

In 68 Se, the systematics of the momentum of inertia suggests that the ground state band is oblate as the band built on top of the 2 + 2 at 1594 keV has a prolate character (see figure 4.4) [START_REF] Fischer | Unravelling the band crossings in 68 Se and 72 Kr : the quest for t = 0 pairing[END_REF]. The B(E2, 2 + 1 →0 + 1 ) was first measured during my PhD [START_REF] Clment | A new device for combined coulomb excitation and isomeric conversion electron spectroscopy with fast fragmentation beams[END_REF] and later with more precision by [START_REF] Obertelli | Shape evolution in self-conjugate nuclei, and the transitional nucleus 68 Se[END_REF]. The value is found compatible with theoretical prediction of an oblate, γ-unstable, deformed state. The spectroscopic quadrupole moment remains un-measured.

In 70 Se, the systematics of the momentum of inertia suggests that the ground state band at low Experimental data are even more scarce in Sr isotopes. Its chemical properties makes Sr element very difficult to extract from an ISOL target. In addition, the N ∼Z 76 Sr isotopes becomes very exotic and produced with very low cross section. Neutron deficient Sr isotopes have a large deformation as deduced from the observation of rather regular rotational ground states band and corresponding large B(E2). In [START_REF] Lemasson | Observation of mutually enhanced collectivity in self-conjugate[END_REF], A. Lemasson et al., have proposed a low mixing between the configurations from the measured B(E2, 2 + →0 + ) placed in the systematics of the correlation between the R 42 =4 + 1 /2 + 1 excitation energy ratio and the B(E2)[W.u.].A -1 . Even if the experimental point follows the systematics of the Kr isotopes, assuming that the R 42 is perturbed by the mixing, a unique feature of 76 Sr arises. Indeed, with a typical mixing strength of 0.2 MeV and a 0 + 2 excitation energy at around 1 MeV, a mixing of 5% is extracted. This apparent contradiction between weak mixing and perturbation of the collectivity calls for more detailed spectroscopic data as the search for the 0 + 2 state and related electromagnetic strength.

The systematics of the 2 + 1 excitation energy and B(E2) for N=Z nuclei are presented in figure 4.5 extracted from [START_REF] Lemasson | Observation of mutually enhanced collectivity in self-conjugate[END_REF]. The systematics show a clear onset of collectivity between 72 Kr and 76 Sr at the N=Z line. It is proposed, at the mean-field level, that an increase of axiallity explains the onset of collectivity [START_REF] Obertelli | Shape evolution in self-conjugate nuclei, and the transitional nucleus 68 Se[END_REF]. A similar systematics of spectroscopic quadrupole moment and quadrupole invariant would confirm the proposed symmetry effect. 4.5 Probing the role of the g 9/2 from laser spectroscopy

One could expect that this increase of collectivity as a function of the proton number could be related to an increase of the g 9/2 occupancy acting as an intruder. Its role can be evaluated from the experimental point of view from the systematics of the spin and parity of the ground states. In figure 4.6, the spin and parity measured for the ground states of odd-masses by laser spectroscopy is presented from Se to Sr. The most probable Nilsson orbital is indicated with the spherical orbit parentage. The deformation character is extracted from laser spectroscopy or deduced from neighbouring even-even nuclei using Coulomb excitation experiment.

In 69 Se (Z=34, N=35), an oblate (from 70 Se shape) 1/2 -spin/parity is assigned. It is proposed that the configuration is dominated by the deformed 1f 5/2 orbit with the 1/2[310] oblate orbital. In 73 Se, the 9/2 + assignment suggests the 9/2[404] orbits, i.e. an oblate deformation, with a g 9/2 orbital acting as an intruder. From Coulomb excitation measurement, 72 Se has a prolate character and the shape inversion to oblate occurs for 70 Se. Therefore, one could assume that 73 Se has a prolate deformation and the local oblate character is surprising. At N=41, the g 9/2 is naturally populated and the 5/2 + spin/parity is compatible with the deformed 5/2[422] orbital. In Br isotopes, with a spin parity 1/2 -and 3/2 -assigned to the 1/2[310] and 3/2[301] deformed f 5/2 orbital, the proton g 9/2 is not yet involved. In Kr isotopes, 73 Kr with a spin/parity 3/2 -points again to the f 5/2 orbital, hence suggesting an oblate shape. Beyond, at N=39 and 41 with a proposed prolate shape as for N=38 and N=40, the 5/2 + spin/parity supports the occupation of the prolate deformed g 9/2 similarly to the Se isotopes. At Z= 37, 75,77 Rb favour also the proton f 5/2 orbital. In Sr isotopes, at N=39, similarly to the Se and Kr isotopes, the ground state suggests the occupation of the neutron deformed g 9/2 . Surprisingly, 79 Sr would correspond to a ground state involving the νf 5/2 orbital.

From the present systematics, the odd-proton isotopes do not point for the occupation of the πg 9/2 as in intruder, at least until Z=37, and underline the role of the deformed πf 5/2 orbitals. In even-proton isotopes, similarly, until N=37, the deformed neutron f 5/2 is favoured. From N=39, 1f 2p 1g the neutron g 9/2 orbital appears as an intruder. But its role in the increase of collectivity when the proton number increases is not established from this systematics. For the Se and Kr isotopic chains, the transition from prolate to oblate deformation when approaching the N=Z line seems to be correlated to the change from the neutron g 9/2 to f 5/2 occupation. 77 Sr and 79 Sr remain prolate with however a change of the neutron orbit. The systematics underline that the shape coexistence, before N=Z, from Se to Sr, seems to be dominated by the competition between the deformed gaps at 34, 36 and 38 in the pf shells. In this section, spectroscopic data such as B(E2) and excitation energy of the first excited states are compared to mean-field calculations using the Gogny D1S or Skyrme SLy6 interactions. This formalism is expected to perform well in these deformed nuclei having complex mixing of configurations. The Se isotopes have been investigated using the Gogny D1S interaction in the mean-field framework in [START_REF] Ljungvall | Shape coexistence in light se isotopes: Evidence for oblate shapes[END_REF]. The known B(E2) are in good agreement with theoretical calculations for the first excited states. These calculations predict a shape change for both 70,72 Se from oblate to prolate at spin 4 and 6 respectively as 68 Se remains oblate as illustrated in figure 4.7. Unfortunately the spin evolution was not probed so far experimentally. Low lying 0 + 2 states are calculated at high excitation energy (≥ 1 MeV). All three 68,70,72 Se are predicted oblate in the 2 + 1 state by the Gogny D1S interaction as a shape change occurs experimentally between 72 Se and 70 Se from prolate to oblate deformation.

In Kr isotopes, the experimental B(E2) and Q s were then compared to mean field calculations using the Gogny D1S or Skyrme SLy6 interactions in [START_REF] Clément | Shape coexistence in neutron-deficient krypton isotopes[END_REF]. In the publication, the later is restricted to axial symmetry. Along the rotational prolate and oblate bands, both calculations reproduce the trend as a function of angular momentum as illustrated in figure 4.8. The Skyrme calculation slightly overestimates the collectivity while the Gogny calculations reproduce well the collectivity as a results of the restriction to axial symmetry in the Skyrme calculation. We proposed that the large overlap between K=0 bands and K=2 γ-band, reflected by large inter-bands B(E2), leads to triaxility and reduces in-band B(E2) and Q s .

In figure 4.9, the experimental level scheme is compared to the Skyrme calculations. As they are restricted to axial symmetry, no K=2 states are calculated. The oblate configuration is predicted to be the ground state by opposition to the experimental conclusions. Finally, inter-band B(E2) are underestimated. Using the Gogny D1S interaction, the agreement is remarkable for excitation energy and matrix elements (see figure 4.10). The K=0 prolate band is predicted to the ground state band as the 0 + 2 and 2 + 3 are part of the oblate rotational band. A K=2 γ band is predicted with a 2 + 2 band head located below the 2 + 3 as experimentally proposed. A strong mixing of K=0 and K=2 components for the 2 + 2 and 2 + 3 states is obtained, as strong indication of triaxility. E0 transition strengths, ρ 2 (E0), are also compared (see figure 4.11). Calculated values are far from In [START_REF] Girod | The role of triaxiality for the coexistence and evolution of shapes in light krypton isotopes[END_REF], potential energy surfaces were published in details following the convincing comparison with the experimental spectroscopic data. As it can be seen, the potential energy surfaces using the Gogny GCM+GOA approach show two minima for prolate and oblate deformation. The J π =0 + 1 projection shows a prolate minima in 76 Kr with an increased spread into the γ triaxial degree of freedom in 74 Kr and more axial oblate ground state in 72 Kr. It was therefore proposed a shape inversion at 74 Kr with a maximum triaxiality as suggested by the experimental data even if the oblate character of the ground state of 72 Kr is not yet firmly established. In the calculations, increasing the angular momentum leads to a narrow minimum in the wave function distribution with less γ-spread and increased axiality. This increased of axiality in the wave function is correlated to the increase of the measured B(E2) therefore reflecting a weaker mixing as proposed from the experimental data. One can finally notice that there is a shape change between spin 2 and 6 in 72 Kr which remains to be proven experimentally. Triaxility seems to be the key ingredient to reproduce the shape coexistence in Kr isotopes. Later on, the Skyrme calculations were developed. Meanfield states generated by triaxial quadrupole constraints that are projected on particle number and angular momentum and mixed by the generator coordinate method on the quadrupole moment were proposed in [START_REF] Bender | Configuration mixing of angular-momentum and particle-number projected triaxial hartree-fock-bogoliubov states using the skyrme energy density functional[END_REF]. This method is equivalent to a seven-dimensional GCM calculations, mixing all five degrees of freedom of the quadrupole operator and the gauge angles for protons and neutrons. The deduced level schemes cancel the previous disagreement. When triaxiality is not authorized, the same results than the axial Skyrme are obtained suggesting that triaxiality seems to be the key to describe prolate-oblate shape coexistence in this region.

Large systematics for the Sr isotopes are not yet available due to the lack of experimental data. The B(E2) and excitation energy of the first 2 + state are well reproduced by the calculations using the Gogny D1S interaction.

Along the N=Z, the calculations reproduce well the onset of collectivity between 68 Se and 76 Sr as shown in figure 4.5. The corresponding potential energy surfaces are also systematically calcu- lated for N=Z nuclei from 60 Zn to 76 Sr in the (β, γ) plan in [START_REF] Girod | The role of triaxiality for the coexistence and evolution of shapes in light krypton isotopes[END_REF]. Figure 4.12 shows an increased of axiallity toward the heaviest masses with a very triaxial and γ-soft minima for 64 Ge, a γ-soft oblate minima in 68 Se, more axial in 72 Kr as the ground state in 76 Sr has a large and axial prolate deformation. This increase of axialilty is associated to the increase of B(E2) experimentally observed and emphasizes the role of the degree of freedom in the mass region [START_REF] Lemasson | Observation of mutually enhanced collectivity in self-conjugate[END_REF].

4.6.2 Shell Model calculations in Se, Kr, Sr and Zr isotopes Mean field calculations have shown a remarkable agreement with experimental data. However, they do not provide a microscopical description. One could wonder for instance, if this clear onset of collectivity associated to axiality between Se and Sr is related to an increase in occupancy of a specific orbital for both proton and neutron. Shell model calculations can provide a more fundamental basis or quantitative influence of a specific orbital or valence space. However, there are severe limitations due to the large valence space needed for such calculations. Monte-Carlo Shell Model Calculations (MCSMC) could in principle be applied like in mass A=100 [START_REF] Singh | Evidence for coexisting shapes through lifetime measurements in 98 Zr[END_REF][START_REF] Togashi | Quantum phase transition in the shape of zr isotopes[END_REF], in the Ni chain [START_REF] Tsunoda | Novel shape evolution in exotic ni isotopes and configuration-dependent shell structure[END_REF] or in the light Hg isotopes [START_REF] Marsh | Characterization of the shape-staggering effect in mercury nuclei[END_REF]. However, this has never been done for N=Z nuclei. ) are compared with the calculations and do not reproduce the onset of collectivity between 72 Kr and 76 Sr. In figure 4.13 taken from [START_REF] Langanke | Shell model monte carlo studies of nuclei in the a 223c80 mass region[END_REF], the g 9/2 occupancy is shown for proton and neutron separately. N=Z nuclei are highlighted with brown squares. The increase of collectivity, in the calculations, from Kr to Zr, is correlated to an increase of the g 9/2 occupancy lot of experimental informations are known (energies, B(E2)'s, g-factors, Qs) and should be an ideal case for SM-like calculations to investigate the microscopical origin of the shape evolution in this mass region. Mean-field calculations work well but only highlight the role of the triaxial degree of freedom. Shell model calculations which should provide a more microscopical description are scare and very limited in their comparison with experimental data but highlight the role of the g 9/2 orbital.

Conclusions

For the future, several systematics are needed :

• The Kr case remains unique a decade after the publication of the present results. Postaccelerated radioactive beam of Se, Sr and possibly Zr are of great interest. Especially 74-70 Se, 72 Kr and 76-80 Sr beams must be Coulomb excited to provide similar set of E2 matrix element as for the Kr case. These will provide a systematic of the shape evolution and mixing along the N=Z line.

• 0 + 2 states remain unknown in 68-70 Se, 76,78 Sr.

• Yrast band's B(E2) have been measured at very low spin up to 72 Kr, 68 Se, 76 Sr.

• no non-yrast E2 matrix elements is known in these isotopes.

• There is a lack of spectroscopic quadrupole moment in the ground state beyond Sr and long-lived isomeric states (particularly 9/2 +m state) in odd-masses by laser spectroscopy.

The electromagnetic probe provides a shape description and possible link between them. However, it does not provide any direct experimental contribution to the microscopical description in particular the role of the g 9/2 orbit. Direct nucleon transfer reactions are the standard technique to identify effective single particle energy and occupancy number. Direct reaction such as (d,p) or (d, 3 He) to probe neutron and proton orbits were never attempted in these isotopes. Furthermore, similarly to the A∼100 region (see chapter II), two nucleon transfer to probe pair excitation origin across the gap 40 are not yet reachable. Two proton transfer from Mo or Zr isotopes to probe the multi pair excitation across Z=40 would have a significant impact in the shape coexistence scenario description. However, the shape identification (prolate or oblate) in the daughter nuclei must be identified first by safe Coulomb excitation. Two neutrons transfer from 74 Kr to 72 Kr would also give more insight in the shape inversion. Single nucleon transfer would be the cornerstone of these studies to tentatively match the shape inversion and mixing with a microscopical description.

Partie V Conclusion Conclusion

The study of the shape coexistence at the 40 sub-shell closure was revolutionized in the last decade thanks to the availability of new instruments, new accelerators, the re-born of relevant experimental techniques and the Monte-Carlo Shell Model Calculations. Shape coexistence remains a spectacular effect in nuclear structure involving strong proton-neutron quadrupole correlation and pairing against the sub-shells gap. The cross comparison between experimental data from in-beam data and laser spectroscopy brings the pieces for a global understanding. Comparison with theoretical calculations remains today partial. Around the 40 sub-shell closure, large set of experimental data can be used as benchmark for large scale comparisons.

In this manuscript, I have reviewed the state-of-the-art of the shape coexistence phenomena involving the sub-shell closure 40 in three mass regions with N=40 around 68 Ni, with Z∼40 in neutron rich isotopes with A∼100 and in self-conjugated nuclei with N∼Z∼40. The two later have benefited in the recent years of experimental results using post-accelerated radioactive ions beam for Coulomb excitation and single nucleon transfer. The Coulomb excitation technique gives access to the spectroscopic quadrupole moment. This is a major step in the description of the shape coexistence in these mass region and reinforces the relevance of the comparison with the most advanced theoretical models. In this manuscript, a particular care was taken in bringing together the different probes such as spin/parity and spectroscopic quadrupole moment from laser spectroscopy in the ground state and the isomers, Coulomb excitation, lifetime measurements, nucleon transfer, E0 spectroscopy and β-decay. The three mass regions have very different level of description.

It is clear that the A∼100 region is characterized by a large amount of experimental data with all techniques showing a clear coherence. Nothing unexpected arise so far. The MCSMC have described for the first time the microscopical mechanism. However, few questions remain. The N=59 isotone evolution as a function of the proton number from Ru to Kr would highlight the role of pairing and proton occupancy. The microscopical description of the Kr isotopes should explain why the offset of deformation is delayed and smoother. Even if the agreement between the MCSMC and some experimental data is remarkable, a global comparison with all spectroscopy data would push forward the constrains to the models. The microscopical description reproduces well the data but this particle-hole scheme coupled to intruder orbital is not experimentally verified. Contrary to the N∼Z nuclei, the agreement between the mean-field calculations and the experimental data is not as good.

In the N∼Z nuclei, large set of experimental data are available for excited states until the Kr isotopes. Clearly, there is a lack of experimental data in the elements from Sr and for all elements in the ground state using laser spectroscopy to draw a global experimental picture. In addition, these nuclei have never been investigated by direct reaction. The agreement between spectroscopic data and the mean-field calculation is remarkable. Microscopical description using conventional or MCSMC have not reached the level of accuracy obtained in the A∼100 region. Further studies using shell models calculations are the next millestone.

Around 68 Ni, the level scheme have been recently clarified. Shell model calculations using conventional approached (LNPS) or Monte-Carlo give a remarkable agreement with the proposed level scheme and first B(E2). The lack of experimental data such spectroscopic quadrupole moments in excited or ground states limits the comparison. The absence of data using laser spectroscopy is evidenced. Similarly to the A∼100 region, the proposed microscopical description is not experimentally verified and systematics studies by direct reaction are needed. In figure 4.15, the archipelago of experimentally proven shape coexistence area is shown. In many aspects, the shape coexistence in A∼100 and around 68 Ni follow the mechanism proposed by K. Heyde and collaborators without any surprises. All new experimental data are successfully compared to state-to-the-art calculations which reinforce the statements of [START_REF] Heyde | Shape coexistence in atomic nuclei[END_REF]. So far, along the nuclear chart, the shape coexistence area can be characterized by this mechanism and experimen-tally proven shape coexistence close to closed shells which do not follow this picture are scarce (see for instance 80 Ge [START_REF] Gottardo | First evidence of shape coexistence in the 78 Ni region: Intruder 0 + 2 state in 80 Ge[END_REF]). However, the N∼Z nuclei around mass A=80 should draw a particular attention. Indeed, as discussed in this manuscript, they do not really follow this description and should attract the future experimental and theoretical works. This universal mechanism was never used to predict new area of shape coexistence. Global calculations in the mean-field approach are available and can be used to predict the next area of shape coexistence. In [START_REF] Möller | Global calculation of nuclear shape isomers[END_REF], P. Möller proposed a global survey of the phenomena. In figure 4.16 from [START_REF] Möller | Global calculation of nuclear shape isomers[END_REF], the number of minima in the potential energy surface is shown for a given nucleus as soon as the depth is greater than 200 keV and its excitation energy lower than 2 MeV. Beyond Z=20, areas with a second minimum overlap very well the figure 4.15. New areas of shape coexistence are predicted in very neutron rich Sn isotopes around N=100 and in super heavy elements around Z=100.

Figure 4.16: The shape coexistence archipelo: calculated number of minima in the potential energy surface of even-even nuclei. Figure from [START_REF] Möller | Global calculation of nuclear shape isomers[END_REF].

Similar comparison can be done using the Gogny D1S interaction using the published calculations from [START_REF] Delaroche | Structure of even-even nuclei using a mapped collective hamiltonian and the d1s gogny interaction[END_REF]. In figure 4.17, stable nuclei are represented with large blue squares. Nuclei indicated with a brown square have a second 0 + state with an excitation energy below 1.0 MeV and a degree of triaxiality lower than γ = 30 degrees. Below Z=28, 0 + 2 states are calculated close to 2 MeV. From Ni isotopes, the calculations overlap very well the figure 4.15. Similarly, next area of shape coexistence in the vicinity of the closed Z=50 and Z=82 proton shells are predicted for very neutron rich isotopes. In the Sn isotopes, 0 + 2 at about 0.9 MeV excitation energy are expected in 146-150 Sn. In Pb isotopes, shape isomer at 0.5 MeV excitation energy are predicted in 224-230 Pb. Finally, several areas are anticipated in the super heavy elements in very neutron rich Fm-No isotopes. In the scheme proposed by K. Heyde, the Sn and Pb isotopes can be anticipated. Indeed, the N=100 shell closure lies between the 1h 9/2 and 2f 5/2 neutron orbital. With proton promoted from the 1g 9/2 to 1g 7/2 , the p-n interaction is similar to the mass A∼100 region. One can also notice than both calculations do not predict an area of shape coexistence near 78 Ni as proposed in 80 Ge [START_REF] Gottardo | First evidence of shape coexistence in the 78 Ni region: Intruder 0 + 2 state in 80 Ge[END_REF]. Figure 4.17: The shape coexistence archipelo: calculated low lying 0 + states in even-even nuclei. Figure from [START_REF] Delaroche | Structure of even-even nuclei using a mapped collective hamiltonian and the d1s gogny interaction[END_REF]. So far the shape coexistence in atomic nuclei was theoretically investigated by effective theories. In a longer perspectives, with the development of Ab -Initio calculations, one could wonder if the phenomenom and its present effective description emerge naturally from the f irst -principles ?

Neutron

Figure 2 . 1 :

 21 Figure 2.1: Systematics of experimental excitation energy of the 2 + 1 state as a function of the neutron number.

Figure 2 . 2 :

 22 Figure 2.2: Systematics of experimental excitation energy of the 2 + 1 states in neutron rich Sr and Sm isotopes as a function of the neutrons added to the shell closure.

Figure 2 . 3 :

 23 Figure 2.3: Systematics of experimental excitation energy of the 0 + 2,3 states as a function of the neutron number.

Figure 2 . 6 :

 26 Figure 2.6: Experimental quadrupole deformation of the ground state for odd-neutron Kr, Sr, Zr, Mo and Ru nuclei, and for the odd-proton Rb, Y, Nb and Tc isotopes, and for the 2 + 1 state in even-even nuclei, as a function of the proton and neutron number. The sizes of the squares are proportional to β 2 under the assumption of axial symmetry. The Pseudo (italic) and Quasi (red) SU(3) orbitals are shown (see text).

Figure 2 . 8 :

 28 Figure 2.8: Systematics of ground and isomeric states spin/parity and confirmed deformation in Zr, Y and Sr isotopes. Orange stars indicate that the deformation parameter was extracted from laser spectroscopy. The yellow stars highlight the deformation measurement by safe Coulomb excitation. The pink stars underline the conclusion from the neutron transfer experiment in 96 Sr. Shapes without star are speculative deformation assignment.

Figure 2 . 9 :

 29 Figure2.9: Shells involved in the mass A=100 region in the spherical shell model. The position of the 2d 3/2 , presented with a dashed line is adopted from the calculations in[START_REF] Togashi | Quantum phase transition in the shape of zr isotopes[END_REF]. The orbitals energy spacing is arbitrary. The microscopical configuration of the spherical ground state of96 Sr is shown.

Figure 2 .

 2 Figure 2.11: Multi-messenger studies of the shape coexistence in 96 Sr

Figure 2 . 12 :

 212 Figure 2.12: Comparison between the experimental and calculated level scheme of 96 Sr using the Gogny D1S interaction. The excitation energies are mentioned in brackets. The arrows are proportional to the B(E2) indicated in e 2 b 2 . The measured and calculated spectroscopic quadrupole moments are given in efm 2 next to each state.

experiment 96 2 (

 962 Sr 5DCH (Gogny) Excited VAMPIR ρ

Figure 3 . 1 :

 31 Figure 3.1: Top panel: Systematics of the experimental energies in the even-N Ca, Ar, S and Si isotopes. Bottom panel: Systematics of the experimental energies in the even-N Fe and Ni isotopes.

Figure 3 . 2 :

 32 Figure 3.2: Level scheme and B(E2) in W.u for the A Co and A+1 Ni isotopes. The member of the 2 + ( A+1 Ni) ×πf -1 7/2 multiplet and the deformed 1/2 -states are shown for the Co isotopes. The first 2 + and proposed deformed 0 + state in Ni are shown.

+ 3 → 2 + 1 )

 21 = 25.3(4)e 2 f m 4 (1.5(2) W.u.) (assuming a pure E2 transition), indicates a single particle character.The position of the proposed 0 + 3 was in a remarkable agreement with the 2202 keV π(2p-2h) 0 +

Figure 3 . 3 :

 33 Figure 3.3: Shape coexistence at N=40. Evolution of the 2p-2h proton intruder and normal νg 9/2 states in 68 Ni, 67 Co and 66 Fe.

Figure 3 . 4 :Figure 3 . 5 :

 3435 Figure 3.4: Shape coexistence at N=40. Microscopical configuration illustration.

Figure 3 . 7 :Figure 3 . 8 :

 3738 Figure 3.7: Multi-messenger approach for the 0 + states in 68 Ni

Figure 4 . 1 :

 41 Figure 4.1: Experimental B(E2↓) in the ground state band of 74,76 Kr from lifetime measurement.

Figure 4 . 4 :

 44 Figure 4.4: Proposed shape assignment in neutron deficient Se isotopes

Figure 4 .

 4 Figure 4.5: B(E2;2 + 1 → 0+ 1 ) systematics at the N=Z line. Figure from [148].

Figure 4 . 6 :

 46 Figure 4.6: Top : Systematics of spin/parity assignment from laser spectroscopy in N∼Z nuclei. The shape mentioned next to the configuration is determined from measured Q s .

4. 6

 6 Short overview of the theoretical approaches 4.6.1 Mean Field calculations in Se, Kr, Sr and Zr isotopes

Figure 4 . 7 :

 47 Figure 4.7: Calculated shape evolution in Se isotopes. Picture from [161].

Figure 4 . 8 :

 48 Figure 4.8: Experimental B(E2,↓) and Q s compared to both Gogny D1S or Skyrme SLy6 calculations.

Figure 4 . 9 :

 49 Figure 4.9: Level Scheme comparison between experimental data and Skyrme SLy6 calculations

Figure 4 . 10 :

 410 Figure 4.10: Level Scheme comparison between experimental data and Gogny D1S calculations

Figure 4 .

 4 Figure 4.11: Theoretical (Gogny) and experimental E0 systematics. Figure from [162].

Figure 4 .

 4 Figure 4.12: HFB potential energy surface for N=Z isotopes. Figure from [162].

Figure 4 . 13 :

 413 Figure 4.13: Theoretical g 9/2 occupancy systematics for Kr, Sr, and Zr isotopes.Figure from[START_REF] Langanke | Shell model monte carlo studies of nuclei in the a 223c80 mass region[END_REF]

Figure from [ 164 ]

 164 Figure 4.13: Theoretical g 9/2 occupancy systematics for Kr, Sr, and Zr isotopes.Figure from[START_REF] Langanke | Shell model monte carlo studies of nuclei in the a 223c80 mass region[END_REF]

Figure 4 . 15 :

 415 Figure 4.15: Experimentally established area of shape coexistence. Figure from [1]

Table 2

 2

	.1: Values of cos 2 θ 0 for the mixing angles between the 0 + states in N=58,60 nuclei extracted
	using the approximation described in the text.
	Elements	N=58	N=60
	Pd	0.93	0.86
	Ru	0.86	0.92
	Mo	0.63	0.84
	Zr	not measured	0.84
	Sr	0.84	0.88
	The cos 2 θ 0 values obtained using the above expression for N=58,60 Sr, Zr, Mo, Ru and Pd
	nuclei are presented in Tab 2.1. For the 0 + 2 → 2 + 1 transition in

  Zr are unknown. In the

	2 β	0.8	Zr Kr Sr	Nb Rb Y
		0.4 0.6	Tc This work gsb Ru this work 2 + 2
		0.2
	-	0.2 0		5/2 d		1/2 s		7/2
	-	0.4
		50	52	54	56	58	60	62
				Neutron number

g Figure 2.5: Quadrupole deformation parameter β 2 as a function of the neutron number in the Kr -Ru isotopic chains. The parameter is calculated assuming an axial symmetry. The values calculated from diagonal E2 matrix elements measured in the present experiment for J =0 states in

g pf ds pseudo SU3 7/2 g 3/2 p 7/2 f 11/2 h Quasi SU3

	Proton Number	38 40 42 44	1/2 s 5/2 d 9/2
		36
		34
			50	52	54	56	58	60	62	64
								Neutron number

Table 3 .

 3

		1: Measured lifetime (τ [ps]) in odd-mass Co isotopes.
			A. Dijon M. Klintefjord V. Modamio B. Olaizola
			[108]	[107]	[110]	[109]
	63 Co	(3/2 -) 995keV	15.4(18)
		(9/2 -) 1383keV	0.9(4)
		(11/2 -) 1674keV		0.55(19)	1.0(3)
	65 Co	(3/2 -) 882keV			5(5)
		(9/2 -) 1480keV (11/2 -) 1480keV (1/2 -) 1095keV	≤17.3 ≤17.3		1.3(5)	1803(28)

Table 3 .

 3

			Assuming Pure E2 Assuming Pure M1
	N=36
	63 Co	(3/2 -) 995keV	3.7(4)	-
		(9/2 -) 1383keV	12(5)	0.013(4)
		(11/2 -) 1674keV	4(1)	-
	64 Ni	2 + 1345.75keV	9.04(28)	-
	N=38
	65 Co	(3/2 -) 882keV	17(16)	-
		(9/2 -) 1480keV	*	*
		(11/2 -) 1480keV	6(2)	-
	66 Ni	2 + 1424.8keV	7.6(13)	-
		(1/2 -) 1095keV	67(1)	1.83(3) × 10 -3

2: Measured B(λµ) [W.u.] in odd-mass Co isotopes and even-even Ni isotopes.

  Ni. The deformed proton

	2 +		+ (0 )	2 + (3025 keV ?)			0 4 +	2 + LPNS = 0.008 0.21(7)	0 3 +	LPNS = 0.9 2.3(2)
				9.04(28)			7.6(13)	3.17(73)
	0	+	64	Ni	0	+	ν g 9/2	66	Ni	+ π 7/2 0 ν g 9/2 f -2	68 Ni
				(1/2 )								1/2
	7/2 11/2 9/2 3/2 4(1)		Co 63 3.7(4) 12(5)	7/2 3/2 11/2 9/2	65	Co	7/2 1/2 m LPNS = 0.02 67(1) 17(16) 6(2) 5(2) 3/2	67 Co	β

Partie IV

Shape coexistence around N=Z=40

Proton number

N = Z l i n e

Ground state

Neutron number 30 spin is oblate and turn to be prolate at higher spin [START_REF] Rainovski | Shape coexistence at high spin in the n = z + 2 nucleus 70 se[END_REF]. Combined Safe Coulomb excitation using post-accelerated beams by A. Hurst [START_REF] Hurst | Measurement of the sign of the spectroscopic quadrupole moment for the 2 + 1 state in 70 Se: No evidence for oblate shape[END_REF] and high precision lifetime measurement by J. Ljungvall [START_REF] Iwasaki | Evolution of collectivity in 72 Kr: Evidence for rapid shape transition[END_REF] have proven the oblate character of the 2 + 1 state. Finally, J. Henderson performed the safe Coulomb excitation of 72 Se [START_REF] Henderson | Localizing the shape transition in neutron-deficient selenium[END_REF]. A negative spectroscopic quadrupole moment for the 2 + 1 state was measured, supporting its prolate character [START_REF] Henderson | Localizing the shape transition in neutron-deficient selenium[END_REF]. Previous measurements in stable Se isotopes by [START_REF] Hayakawa | Projectile coulomb excitation of 78 Se[END_REF][START_REF] Kavka | Coulomb excitation of 76,80,82se[END_REF] have also proved a prolate 2 + 1 state. There is therefore a shape change for the 2 + from prolate to oblate in 70 Se. No spectroscopic quadrupole moment are known beyond the 2 + 1 state. Recently, the first spectroscopy of 70 Kr was performed and suggests, by comparison with theoretical calculations, a shape coexistence scenario [START_REF] Wimmer | Shape coexistence and isospin symmetry in a 20ac= 20ac70 nuclei: Spectroscopy of the tz 20ac= 20ac 02c6 20191 nucleus 70kr[END_REF]. In 72 Kr, using a comparison between measured B(E2; 2 + → 0 + ) and theoretical values, Iwasaki et al [START_REF] Iwasaki | Evolution of collectivity in 72 Kr: Evidence for rapid shape transition[END_REF] proposed that the first 2 + state in 72 Kr is compatible with an oblate shape. The safe Coulomb excitation of 74,76 Kr provides the first evidence for a shape coexistence scenario with a prolate ground state band and an oblate for both proton and neutron. In the Kr chain, the neutron g 9/2 occupancy increases with mass as expected with a calculated shape change between 72 Kr and 76 Kr. The increased number of neutron in the g 9/2 favours the prolate deformation. The f 5/2 p 1/2 g 9/2 d 5/2 occupancies, assuming that there are equal for proton and neutron, are shown for 68 Se, 72 Kr and 76 Sr in figure 4.14. It is known experimentally that 68 Se has a lower collectivity [START_REF] Clment | A new device for combined coulomb excitation and isomeric conversion electron spectroscopy with fast fragmentation beams[END_REF][START_REF] Obertelli | Shape evolution in self-conjugate nuclei, and the transitional nucleus 68 Se[END_REF]. In the calculations for 68 Se, the g 9/2 d 5/2 occupancy is very low (∼ 0.2) for the 0 + 1,2 and 2 + 1 states, ie equivalent for both ground states band and excited configuration. One could speculate that this similarity favours large mixing and consequently lower collectivity. For 72 Kr and 76 Sr, the f 5/2 occupancy increases as expected. The g 9/2 d 5/2 occupancy increases between 68 Se and ( 72 Kr, 76 Sr) but remains still low (∼ 1 particle) for the ground state band and ∼ 0.5 particle for the 0 + 2 . Contrary to 68 Se, there is a clear difference between the ground states band and the excited configuration which could reduce the mixing inducing a more clear shape coexistence scenario with two 0 + states. The increase by a factor 2 of the experimental B(E2)'s between 68 Se and 72 Kr is reproduced supporting the contribution of the g 9/2 orbital. Experimentally, there is a second step by a factor 2 between 72 Kr and 76 Sr in B(E2) which is also reproduced in the calculation associated to a small increase of the g 9/2 occupation (∼ 0.5 particle).

More recent shell model calculations using the JUN45 and the LNPS interactions have been performed by B. Bounthong in [START_REF] Bounthong | Calculs microscopiques pour les noyaux exotiques de masse moyenne et lourde[END_REF]. In the calculations, the B(E2; 2 + 1 → 0 + 1 ) increases by only 20% between 72 Kr and 76 Sr as the g 9/2 occupation remains almost identical, supporting again the role of this orbital in the collectivity of the N=Z nuclei.

In summary, some shell model calculations have investigated the 2 + state in N=Z nuclei and in the Kr, Sr and Zr chain leading to incomplete conclusions. The Kr chain remains the case where