Nuclear-structure studies of exotic nuclei with MINIBALL

Due to the requirements needed for low-intensity radioactive ion-beam experiments at significant recoil velocities the MINIBALL array was developed to obtain high-energy resolution and high detection efficiency for nuclear reactions with low γ-ray multiplicity events. The γ-ray spectrometer is built of large volume n-type high-purity germanium (HPGe) crystals (length 78 mm, diameter 70 mm) which are individually encapsulated. The outer contact of the hexagonal shaped crystal is electronically six-fold segmented. Three of these crystals are assembled together in a common cryostat forming a triple-cluster detector. Eight triple-cluster detectors are installed on arc-shaped arms which allow a flexible mounting of the HPGe detectors around the target chamber. In most cases a high solid-angle coverage is accomplished in a compact configuration. With the optimum distance between target chamber and Ge-detectors, the array covers a solid angle of about 60% of 4π. Due to its flexibility special requirements of detector configurations are easily accommodated e.g. for measurements of angular correlations, nuclear g-factors or lifetimes.

Front-end signal processing of the 168 individual signals from the central core electrodes and segment contacts is based on broad-band preamplifiers with cold FET technology and digital electronics, enabling count rates of up to 20 kHz per detector. An average energy resolution of 2.3 keV at $E=1.3\,\mathrm{MeV}$ is achieved. The photopeak efficiency of MINIBALL is about 8% at this γ-ray energy for the compact configuration. The electrical segmentation of the outer detector contact increases the granularity and thus reduces the effective solid angle for the individual γ-ray. In this way large Doppler shifts and broadening are corrected effectively for in-flight γ-ray emission at high velocities even at the very close distances between target and detector. In addition pulse shape analysis of all detector signals is possible. This allows the first interaction point of the γ-ray to be defined in order to improve the position resolution. The combined information from the charge collection at the central electrode and at the segments gives the interaction point of the γ ray in radial and azimuthal direction. A two dimensional position resolution of about 7.5 mm (FWHM) is obtained with the segmented MINIBALL HPGe detectors. A detailed description of the spectrometer is provided by [51–54]. A picture of the spectrometer and its surrounding infrastructure is shown in figure 1.

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To ensure a proper Doppler correction for in-flight γ-ray emission at high velocities, the angular information of the γ ray, provided by the position-sensitive MINIBALL detectors, is combined with the direction and velocity of the scattered beam particle that was measured in coincidence. For this purpose the scattered beam and recoiling target nuclei are detected by silicon strip detectors which are employed in two different configurations for identification and reconstruction of the trajectories of the scattered nuclei. First, a CD-shaped double-sided silicon strip detector (DSSSD) is typically used for CE experiments in order to detect forward-scattered beam- or target-like reaction products. This device consists of four identical quadrants with different Si wafer thicknesses between 35 and 500 μm. Each quadrant is highly segmented by 16 annular strips of 2 mm pitch at the front side and by 24 radial strips of ${3.5}^{\circ }$ pitch at the back side. The detector geometry covers the angular range between ${16}^{\circ }$ and 54${}^{\circ }$ under forward direction [55].

A second particle detector array was realized later in 2007 in order to extend the physics program at REX-ISOLDE to transfer reactions in inverse kinematics. The T-REX target chamber was designed and was built for maximum efficacy for both particle and γ-ray detection. The T-REX array consists of two CD-shaped 500 μm DSSSDs, one in forward and one in backward direction, and a barrel of eight position-sensitive planar detectors, positioned symmetrically around ${90}^{\circ }$. Both the forward and backward DSSSDs are stacked with 500 μm unsegmented E-detectors in order to form ${\rm{\Delta }}E-E$ telescopes for identification of different reaction products. These can be the different light ejectiles from the target (p, d, t, α) and heavier isotopes, such as beam-like particles or the heavy carrier material of the target in forward direction. The forward and backward barrel detectors are identical and consist of a stack of 140 and 1000 μm planar Si detectors. The thinner Si detectors are segmented in 16 strips perpendicular to the beam axis. Position information along these strips is obtained from the charge division on a resistive layer. The thicker Si detectors are not segmented. This stack of Si detectors serves also as a ${\rm{\Delta }}E-E$ telescope to identify the different ejectiles from the target. All detectors are protected by foils, which stops beam particles scattered off the target. The energy resolution for light particles ranges typically from 250 keV to 2 MeV, depending on their scattering angle, energy and the target thickness. Therefore, it is crucial to detect coincident γ rays, which are used to identify the populated final state with the much higher HPGe-detector resolution. By gating on a γ ray which depopulates the state, it is possible to deduce the angular distribution of the emitted protons which correspond to the excitation of that state. The angular resolution of the particle detector is typically better than ${5}^{\circ }$ [56, 57].

A position-sensitive parallel plate avalanche counter can be placed on the beam axis behind the target for measurement of the beam profile perpendicular to the beam direction. It consists of a central anode foil and two segmented cathode foils, mounted inside a gas volume. Both cathode foils are segmented to allow a position sensitive readout of the particle flux in both directions.

Close to the beam dump position a Bragg ionization chamber is mounted, to allow a permanent monitoring of the beam composition in mass and charge. A homogeneous electrical field can be applied along the gas volume on 20 electrodes. The incoming ions loose their energy in the gas, following the characteristic energy loss curve of the Bethe-Bloch formula. Due to the applied electrical field the generated charges drift with a constant velocity towards the anode. From the time evolution of the anode pulse the mass A and charge Z of the incoming particle is deduced.

The device that is commonly used to continuously survey the RIB composition is a ${\rm{\Delta }}E-{E}_{{\rm{Res}}}$ detector. It consists of a gas-filled ionization chamber with CF₄ at a pressure of 300–450 mbar, generating a charge Z dependent energy loss ${\rm{\Delta }}E$ signal, coupled to a silicon detector for the measurement of the residual energy E_Res. As all ions have the same energy per nucleon, the mass A can be derived from E_Res. An adjustable collimator is put in front of the entrance windows of the ionization chamber to attenuate the incoming beam intensity in order to prevent pile-up.

The radioactive ion beam provided by REX-ISOLDE is not a continuous flux of particles, but it is delivered in short particle bunches with a characteristic time structure. It is caused by the time-dependent yield of the ions of interest due to their release and radioactive decay and by the duty cycles of the charge-breeding and accelerating systems. While the small duty cycle can aggravate the effect of background arising from pile-up and random coincidences, in-beam spectroscopy experiments can benefit from this feature by efficient background suppression. In order to synchronize the data acquisition at the MINIBALL setup with the production and acceleration of the radioactive ions, different time signals are exploited at ISOLDE. The first periodic time signal is the time pulse at the beginning of each super cycle of the CERN PS Booster (PSB). This time signal is especially relevant for experiments with laser ion sources, to drive a shutter which periodically blocks the laser light towards the ionization tube. The analysis of the nuclear reaction events can be limited to a time window which combines various conditions, e.g. (1) the time difference of a few half-lifes of the ions of interest with respect to the signal of proton beam impact can be selected; (2) there can be a requirement that the shutter of the laser ion source is open, allowing the element of interest to be selected; (3) a time window can be employed which signals that the charge bred ions are injected from the EBIS into the accelerator. In this way background from long-lived or stable contaminants can be strongly suppressed.

The exotic neutron-rich nuclei at N = 20 are up to now one of the prime examples where unexpected sudden changes of nuclear properties, associated with the traditional landmarks of nuclear structure, occur. For these nuclei it has been realized that the interaction of the last valence protons and neutrons, in particular the monopole component of the residual interaction between those nucleons, can lead to significant shifts in the single-particle energies, leading to the collapse of classic shell closures and the appearance of new shell gaps. The island of inversion around ³²Mg, with neutron-rich nuclei at and close to the magic number N = 20 (illustrated in figure 2) shows strongly deformed ground states in Ne, Na, and Mg isotopes. This is related to the reduction of the N = 20 shell gap and quadrupole correlations enabling low-lying deformed $2p2h$ intruder states from the higher fp shell to compete with spherical normal neutron $0p0h$ states of the sd-shell. The promotion of a neutron pair across the N = 20 gap can result in deformed intruder ground states and the two competing configurations can lead to the coexistence of spherical and deformed ${0}^{+}$ states in the neutron rich nuclei ${}^{\mathrm{30,32}}$Mg.

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The first convincing indication for the low-lying intruder states was deduced from the ${2}^{+}$ state measured in RIKEN in 1995 for ³²Mg using the new technique of intermediate-energy CE [58]. A large $B(E2)$ value of $454(78){{e}}^{2}{{fm}}^{4}$ was measured corresponding to a large quadrupole deformation of the ground state. These findings were exceeding expectations for a semi-magic nucleus. Later on the $B(E2)$ values for the neutron-rich Mg isotopes ${}^{\mathrm{30,32,34}}$Mg were measured by different groups at the intermediate-energy facilities MSU [59, 60] and GANIL [61] showing a wide range of results (see figure 3).

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In order to clarify the situation a spectroscopic program with MINIBALL was started in 2003 and transition-strength values in ${}^{\mathrm{30,31,32}}$Mg and in ${}^{\mathrm{29,30}}$Na were determined employing the complementary technique of sub-barrier CE in inverse kinematics at safe energies. In comparison with the intermediate-energy CE, the new approach gave improved γ-ray energy resolution and cleaner background conditions. The results are barely influenced by nuclear interference effects and by excitations of higher-lying states. The Radioactive beam EXperiment for the post-acceleration of radioactive beams was ideally suited for this purpose. The maximum beam energy was up to 3.0 MeV/A at this time. De-excitation γ rays were detected by the MINIBALL spectrometer in coincidence with scattered particles in the CD shaped segmented Si-detector under forward direction.

The CE experiments have proven that the deformed sd − pf intruder configuration is dominating the ground state of ³²Mg [63]. In contrast a much reduced excitation of $B(E2)=280(20){e}^{2}{{fm}}^{4}$ for the ${2}^{+}$ state at 1483 keV was found in the N = 18 isotope ³⁰Mg [11], which is consistent with a spherical ground state configuration. The clear experimental results from the two even–even isotopes ${}^{\mathrm{30,32}}$Mg inspired a detailed investigation of the unstable odd-N nucleus ³¹Mg. The ground state properties of ³¹Mg indicated already a change of nuclear shape at N = 19 with a deformed ${J}^{\pi }=1/{2}^{+}$ intruder state as a ground state, implying that ³¹Mg is part of the island of inversion [68].

Therefore, the collective properties of excited states in ³¹Mg were examined by means of CE. The level scheme of ³¹Mg was extended. Spin and parity assignment of a new state at 945 keV yielded $5/{2}^{+}$ and its de-excitation is dominated by a strong collective M1 transition. The properties of a positive-parity yrast band with $K=1/2$, built on the $1/{2}^{+}$ ground state, are in good agreement with the $5/{2}^{+}$ state at 945 keV. The measured $B(E2)$ and $B(M1)$ values support this assumption. The increased collectivity is well described by the deformed Nilsson model for excited states in ³¹Mg. Finally, the quadrupole moment supports the idea that for the N = 19 magnesium isotope not only the ground state but also excited states are no longer dominated by a spherical configuration. Although the E2 strength of the $5/{2}^{+}\to 1/{2}^{+}$ transition in ³¹Mg is smaller it is comparable to the corresponding ${2}^{+}\to {0}^{+}$ transition in ³²Mg. The deviation could be explained by the assumption that higher-lying states in ³¹Mg are not dominated by pure intruder configurations, but contain an admixture of other particle-hole configurations [13]. The measured E2 matrix elements along the Mg isotopic chain is given in figure 3 illustrating the contribution of the safe CE results in comparison with intermediate-energy CE and a complementary fast-timing experiment from ISOLDE by Mach.

Nuclear shell evolution in neighboring neutron-rich Na nuclei at N = 20 was studied by determining reduced transition probabilities, in order to map the border of the island of inversion. To this end CE experiments, employing radioactive ${}^{\mathrm{29,30}}$Na beams with a final beam energy of 2.85 MeV/A, were performed employing the MINIBALL spectrometer in coincidence with scattered particles. Transition probabilities to excited states were deduced. The measured B(E2) values agree well with shell-model predictions, supporting the idea of a gradual transition in the Na isotopic chain. While in ²⁹Na the ground-state wave function contains significant intruder admixture already at N = 18, results for ³⁰Na with N = 19 are consistent with an almost pure $2p2h$ deformed ground-state configuration as predicted by the Monte Carlo shell-model (MCSM) calculations using the SDPF-M interaction [72]. For ²⁹Na a mixing of intruder configurations by 42% and 32% is calculated for the wave function of the $3/{2}^{+}$ ground state and the first excited $5/{2}_{1}^{+}$ state, respectively. The onset of large intruder admixtures in the ground state wave function was established already for the N = 18 isotope [14]. A comparison between experimental results from MSU, TRIUMF and REX ISOLDE and the MCSM calculation is given in figure 4. Figure 4 includes also a recent E2 matrix element value extracted from a ground state transition in ²⁶Na, where no intruder configuration is relevant.

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The CE experiments described above focused on the intruder states in the island of inversion with configurations consisting of two neutrons excited from the $0{d}_{3/2}$ and $1{s}_{1/2}$ orbitals into the $0{f}_{7/2}$ and $1{p}_{3/2}$ orbitals across the N = 20 shell gap, forming a $2p2h$ state. A complementary MINIBALL experiment provided the first confirmation that also the predicted coexisting excited state similar to the closed-shell configuration exists. The beam energy at the REX-ISOLDE facility also enabled transfer-reaction experiments to be performed. Therefore, ³²Mg was studied by a (t, p) two-neutron transfer reaction in inverse kinematics with a ³⁰Mg beam. This involved the use of a radioactive tritium target in combination with a radioactive heavy-ion beam. Light charged particles emitted from the target were detected and identified by the T-REX particle detector while γ rays were detected by the MINIBALL. The angular distribution of the protons allowed the angular momentum transfer ΔL of the reaction to be determined. The analysis of excitation energies and angular distribution of the protons led to the first observation of the excited shape-coexisting ${0}_{2}^{+}$ state in ³²Mg at 1058(2) keV. A more detailed description of these results is given in the chapter about nuclear reactions in this special edition volume.

The re-accelerated radioactive ion beams from REX-ISOLDE were of great relevance for nuclear-structure investigations in very neutron-rich isotopes in the vicinity of ⁶⁸Ni and ⁷⁸Ni, spanning the range between the shell gaps at neutron numbers N = 40 up to N = 50. The experimental campaign started out more than ten years ago when first predictions related to the evolution of shell structure where made and the important role of the tensor interaction and three-body forces were awaiting experimental verification.

At this point an experimental program at REX-ISOLDE commenced with a series of ground-breaking CE measurements which established not only a whole series of new transition-strength values but explored in several cases new and unknown states in the most exotic Cu and Zn isotopes. From the experimental point of view two major advances for in-beam γ-ray spectrocopy with ISOL beams were pioneered in this mass region. First, the use of re-accelerated isomeric radioactive ion-beams in Cu isotopes was established. The isomers were selectively extracted from the ion source by the resonant-ionization laser technique. Second, the sophisticated coupling of the efficient REX-TRAP and the REX-EBIS together with the exploitation of different time sequences for isotopes with different lifetimes allowed beams of refractory Fe isotopes to be prepared. In this way a major obstacle and limitation of the ISOL technique was overcome.

Transfer reactions gave access to the direct neighbors of ⁶⁸Ni at N = 40 and ⁷⁹Zn at N = 50 providing an excellent tool to probe the size of the shell gaps and test the single-particle character of unknown neutron orbitals. One of the first g-factor measurements with a radioactive ion beam was successfully performed in ⁷²Zn using the low-velocity transient-field technique [24].

3.2.1. Coulomb excitation of refractory Fe isotopes

3.2.2. The case of ${}^{68}{\rm{Ni}}$

3.2.3. Isomeric beams of Cu isotopes

3.2.4. Neutron-rich Zn isotopes and the N = 50 shell closure

The isotopes between ¹⁰⁰Sn and ¹³²Sn constitute the longest semi-magic chain in the nuclear chart and they have for this reason since long been a testing ground for nuclear models. The purpose of the REX-ISOLDE experiments in the ¹⁰⁰Sn region has sofar been to investigate the robustness of the N = Z = 50 shell closure using CE. Several isotopes, namely ${}^{\mathrm{100,102,104}}$Cd [31], ${}^{\mathrm{107,109}}$Sn [33, 34] and ${}^{\mathrm{106,107,108}}$In [32, 35], have been studied with measurements on ${}^{\mathrm{106,108,110}}$Sn as highlights [29, 30] (see figure 5).

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In the experiments on the even ${}^{\mathrm{100,102,104}}$Cd isotopes Ekstrom et al [31] extracted data for the reduced transition probability, B(E2; ${0}_{{gs}}^{+}\to {2}_{1}^{+}$), and the quadrupole moment, Q(2${{}_{1}}^{+}$), and developed a maxium likelihood method for the probability distribution of the (B(E2),Q(2${{}_{1}}^{+}$)) pair. The analysis method, applied for the first time in a radioactive beam experiment, was used to investigate the neutron effective charge in the ¹⁰⁰Sn region. The experiment also put a limit on the B(E2; ${0}_{{gs}}^{+}\to {2}_{1}^{+}$) in ¹⁰⁰Cd. In measurements on ${}^{\mathrm{106,108}}$In, Ekstrom et al [32] reanalyzed the $\pi {g}_{9/2}^{-1}\otimes \nu {d}_{5/2}$ and $\pi {g}_{9/2}^{-1}\otimes \nu {g}_{7/2}$ multiplets. They also confirmed the 7⁺ spin and parity of the ground state in ¹⁰⁶In and concluded that the ordering of the ground state and close lying isomeric state had inverse order compared to shell model predictions in ¹⁰⁸In. In experiments on the light odd Sn isotopes, Di Julio et al [33, 34] investigated the structure of ${}^{\mathrm{107,109}}$Sn and put constraints on the single-particle dominated orbits in the odd Sn isotopes. In experiments on ${}^{\mathrm{106,108,110}}$Sn Cederkall et al [29] and Ekstrom et al [30] addressed the strength of the shell closure at ¹⁰⁰Sn. The remaining part of this section will expand on this topic.

In the even Sn isotopes the wave function of the lowest excited states is to a large extent built on the $\nu ({d}_{5/2},{g}_{7/2},{s}_{1/2},{d}_{3/2},{h}_{11/2})$ orbits above the N = 50 gap. On the neutron deficient side, i.e. for Sn isotopes with a mass number, A < 112, the $\nu ({d}_{5/2},{g}_{7/2})$ orbits should dominate if ¹⁰⁰Sn is a good shell closure. Potential components e.g. from excitations across the Z = 50 gap that would change this picture can be investigated using different methods. The approach used at REX-ISOLDE relies on determining the reduced transitions probabilities from the ground state to the first ${2}^{+}$ state, the B(E2; ${0}_{{gs}}^{+}\to {2}_{1}^{+}$), in even Sn isotopes as function of neutron number. The results are then compared to predictions from large scale shell model calculations (LSSM) based on the assumption that ¹⁰⁰Sn is doubly magic.

3.3.1. The light even Sn isotopes

When the program started in 2003 there was an indication of a deviation between prediction and experiment in the B(E2; ${0}_{{gs}}^{+}\to {2}_{1}^{+}$) for ${}^{\mathrm{112,114}}$Sn, while the even isotopes from ¹¹⁶Sn to ¹³⁰Sn were well described by LSSM calculations (see figure 6). In parallel with the preparations at REX-ISOLDE, a complementary experiment, aiming to use a relativistic beam of ¹⁰⁸Sn, was also in preparation at GSI. The result from that measurement, which was published in 2005 [76], pointed to a difference between the experimental B(E2; ${0}_{{gs}}^{+}\to {2}_{1}^{+}$) value and model predictions also for that isotope but it had a large statistical error.

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The need to use radioactive ion beams to expand the B(E2; ${0}_{{gs}}^{+}\to {2}_{1}^{+}$) measurements to neutron deficient Sn isotopes stems from isomerism in the yrast cascade which hampers measurements of the reduced transition probabilities of the first ${2}^{+}$ state using fusion evaporation reactions and traditional Doppler shift methods. This isomerism arises from the relatively small energy difference between the yrast ${4}^{+}$ and ${6}^{+}$ states, where the ${6}^{+}$ state is the highest-spin state that can be constructed by breaking a pair in the $\nu ({d}_{5/2},{g}_{7/2})$ dominated ${0}^{+}$ ground state. In addition, the use of CE at safe energies has the advantage of producing results that can be compared to measurements of the stable isotopes in the chain without the risk of introducing method dependent systematic differences.

Different target materials were investigated as the first step of the program. Data from ORNL and GSI [77] showed that molecular beams of SnS from a Ce₂S_x target could be advantageous in order to suppress isobaric contaminants using the molecular sideband. This approach should also increase the intensity for beams of short-lived Sn isotopes due to the relatively short release times for the SnS molecule. Desorption times two orders of magnitude shorter for SnS compared to atomic Sn has been reported the literature [77]. Target tests were therefore performed to benchmark the molecular SnS beam against laser ionization using an LaC_x target. The intensity of the laser ionized beam proved to be considerably higher than for the molecular sideband. Although significant amounts of In isobars, from surface ionization, could be expected due to the relatively low ionization potential of In (5.78 eV), it was concluded that laser ionization could be employed for Sn beams down to mass number 106, under the condition that the ionization efficiency was monitored during the experiment to determine beam purity. The lifetimes of ¹¹⁰Sn, ¹⁰⁸Sn, ¹⁰⁶Sn are 4.1 h, 10.3 m and 115 s, respectively and the release time of Sn from a LaC_x matrix at ca 2000 degrees is 100 s or longer [77]. Sn beams extracted from an LaC_x target using laser ionization are therefore, with a typical PS booster supercycle, almost continuous. In contrast, the contaminant In is released within seconds. Yield measurements show that ca 85% of the In isotopes with half lifes above 5 s exit the target before they decay [78]. The amount of contaminant could therefore be reduced by using a supercycle with consecutive pulses followed by several empty pulses. This method gave a beam purity of ∼90%, 60% and 30% for ¹¹⁰Sn, ¹⁰⁸Sn and ¹⁰⁶Sn, respectively.

Four runs were performed using post-accelerated beams of ${}^{\mathrm{110,108,106}}$Sn at an energy of 2.8 MeV A^–1. The trapping efficiency was 40%–60% while the charge breeding, to the 26⁺, 27⁺ and 30⁺ charge states used in the different experiments, had typical efficiencies of 8%–10%. The transmission through the post-accelerator varied between 60% and 80% thus yielding a typical total transmission of ca 5% from production to secondary target. The intensities at the secondary target were 10⁶ p s^–1 for ${}^{\mathrm{110,108}}$Sn and 10⁵ p s^–1 for ¹⁰⁶Sn. For ¹¹⁰Sn the intensity of the proton driver was reduced to 7 × 10¹² p/pulse compared to the nominal 3 × 10¹³ p/pulse in order not to overwhelm the detector system. Following bombardment of targets of ⁵⁸Ni with typical thickness of 2 mg cm⁻², γ-ray yields of 588(24), 994(38) and 133(14) counts were obtained for the ${2}_{1}^{+}\to {0}_{{gs}}^{+}$ transition in respectively ¹¹⁰Sn, ¹⁰⁸Sn and ¹⁰⁶Sn. The corresponding B(E2; ${0}_{{gs}}^{+}\to {2}_{1}^{+}$) values were determined to be 0.220(22) ${e}^{2}{b}^{2}$, 0.222(19) ${e}^{2}{b}^{2}$ and 0.195(39) ${e}^{2}{b}^{2}$.

These results, which were pioneering measurements of the reduced transition probabilites in the ¹⁰⁰Sn region, remain unique. They have considerably better precision than the earlier measurement on ¹⁰⁸Sn and therefore stand as the first systematic confirmation that current LSSM calculations cannot properly account for the trend of the B(E2; ${0}_{{gs}}^{+}\to {2}_{1}^{+}$) values in the Sn chain. The measurements at REX-ISOLDE were followed by measurements using high-energy CE at NSCL(${}^{\mathrm{104,106},\mathrm{108,110}}$Sn), GSI (¹⁰⁴Sn), and at RIKEN (¹⁰⁴Sn). Taken together these measurements confirm a stronger excitation strength to the first ${2}^{+}$ state than expected from LSSM calculations (see figure 6) although some deviation can be seen for the GSI measurement of ¹⁰⁴Sn, which was limited by statistics.

Several calculations were performed as part of the interpretation of the REX-ISOLDE experiments in the ¹⁰⁰Sn region. From the theoretical standpoint the Sn chain has been a good example of the seniority scheme [86] exhibiting almost constant ${2}_{1}^{+}$ energies and an increasing trend of B(E2; ${0}_{{gs}}^{+}\to {2}_{1}^{+}$) values from the N = 82 shell closure towards midshell. Such simple relations are not valid per default in general seniority, although Morales et al [87] have shown that simple trends can be reproduced under certain conditions. Studies of the overlap of the seniority-two wave function with the first 2⁺ state in the shell model, e.g. by Sandulescu et al [88], give overlaps ranging from ca 93% in ¹⁰⁴Sn down to ca 42% in ¹¹²Sn. This indicates that a trunctation scheme based on seniority-two states is inadequate for heavier isotopes in the Sn chain.

The general dependence on neutron number sketched above is nevertheless reproduced in many LSSM calculations. A set of illustrative calculations performed by Banu et al [76], and later expanded on by Ekstrom et al [30], are good benchmarks for LSSM calculations in the light Sn isotopes and describe the current limitations. The first of these using a ¹⁰⁰Sn core is presented as a typical result of an LSSM calculation in figure 6. It is in excellent agreement with the adopted values for the stable isotopes and the ORNL data for the heavy unstable isotopes, while it underestimates the transition strengths below ¹¹⁴Sn. To investigate the effects of proton excitations across the Z = 50 gap the same authors performed a calculation including up to 4p–4h proton core excitations. This calculation, which was performed with seniority truncation, increases the transition strength midshell but does not reproduce the strength for the lower mass numbers. Ekstrom et al [30] performed additional calculations, using a ¹⁰⁰Sn core, and included charge symmetry breaking as well as charge independence breaking. Their calculation shows a first sign of an asymmetric trend of the reduced transition probabilities with respect to midshell but does still not produce the enhancement for the lower mass numbers. Additional work has also been done by Ansari [82] using RQRPA (see RQRPA label in figure 6). Although the prediction underestimates the transition strength in the heavy isotopes a clear increase in strength, reproducing the general behavior of the experimental data, can be seen for the lighter isotopes. This prediction prompted remeasurements of the B(E2; ${0}_{{gs}}^{+}\to {2}_{1}^{+}$) in the midshell isotopes using Doppler shift attenuation methods at GSI and ANU. Interestingly, those results show statistically significant deviations from the adopted values for the same isotopes. This situation still remains and new measurements are considered.

Finally, the basic input parameters to any shell-model calculation are the two body matrix elements (TBME) and the single particle energies (SPE) with respect to the core. As the structure of ¹⁰¹Sn has still not been established in experiment, calculations continue to rely on SPEs and TBMEs extracted from excitation spectra of neighboring atomic nuclei and, for TBMEs, on derivations based on microscopic descriptions of the nucleon–nucleon interaction. The microscopic approach to TBMEs is attractive as it provides a way to test microscopic descriptions of the nucleon–nucleon interaction in few-particle nuclei [89]. However, as the SPEs in the Sn region are largely unknown [90, 91] one aim for the future program at HIE-ISOLDE is therefore to perform transfer reactions to the single-particle dominated states in the odd Sn isotopes to cast further light on this topic.

Bauer et al [44] and Stegmann et al [45] investigated the evolution of quadrupole collectivity for the N = 80 isotones, by measuring the $B(E2;{2}_{1}^{+}\to {0}_{1}^{+})$ in ¹⁴⁰Nd and ¹⁴²Sm respectively. While the value of $B(E2)$ for ${}_{60}^{140}$Nd has the expected dependence on ${N}_{\pi }{N}_{\nu }$ (whereas a dip is observed [92] for ${}_{58}^{138}$Ce corresponding to the $\pi ({g}_{7/2})$ subshell closure at Z = 58), the value for ${}_{62}^{142}$Sm deviates significantly from the ${N}_{\pi }{N}_{\nu }$ behavior, reflecting the impact of the $\pi (1{g}_{7/2}2{d}_{5/2})$ subshell closure at Z = 64.

Grahn et al [93] measured the $B(E2;{2}_{1}^{+}\to {0}_{1}^{+})$ in ²⁰⁶Po and ${}^{\mathrm{208,210}}$Rn. They observed that contributions from collective excitations are present in these nuclei when moving away from the N = 126 closed shell, although they have been proposed to lie in, or at the boundary of the region where the seniority scheme should persist.

Following the observation of low-lying ${0}^{+}$ states in the light Kr isotopes Clément et al [97] carried out pioneering experiments, in this case using the SPIRAL facility at GANIL, to accelerate ⁷⁴Kr and ⁷⁶Kr isotopes, enabling the CE yields of several states in these nuclei to be measured. By combining the Coulex data with nuclear lifetime date, enough matrix elements could be measured accurately to enable the shape of the ground-state band and the band built on the excited ${0}^{+}$ state to be determined in a model-independent way using rotational invariant products of the E2 operator. This provided firm experimental proof that the ground states of both nuclei are prolate, while the excited ${0}^{+}$ state bands are triaxial. Using the REX-ISOLDE facility Hurst et al [37] measured the Coulex yield of the lowest ${2}^{+}$ state in ⁷⁰Se. In order to remove isobaric contaminants, the ⁷⁰Se was extracted from the ZrO₂ production target in the form of a complex molecule (⁷⁰Se¹²C¹⁶O⁺) that was broken up in REXEBIS, so that a two-stage A/q selection could be applied. By combining the Coulex yield with the available lifetime measurement a prolate shape for the ${2}^{+}$ state was deduced. A subsequent re-measurement of the ${2}^{+}$ lifetime found it to be much longer [98], so that only the hypothesis of an oblate shape is consistent with both Coulex yield and lifetime, in agreement with theoretical expectations.

The sudden onset of deformation observed for neutron rich Zr and Sr nuclei at N = 60 belongs to the most dramatic shape changes in the nuclear chart. This shape transition is accompanied by the appearance of low-lying ${0}^{+}$ states that, for $N\lt 60$, can be interpreted as a deformed configuration that becomes the ground state at N = 60, while the spherical ground-state configuration of the isotopes with $N\lt 60$ becomes nonyrast for those with $N\geqslant 60$ (see [95] and references therein). The systematics of the excitation energies of the ${2}_{1}^{+}$ states and $B(E2;{0}_{1}^{+}\to {2}_{1}^{+})$ for even–even nuclei around N = 60 are given in figure 7. The shape transition is explained (see [38] and references therein) as a consequence of the strongly interacting proton and neutron Nilsson orbitals. For the neutrons, the downsloping $\nu 1/{2}^{-}[550]$ and the $\nu 3/{2}^{-}[541]$ orbitals, both resulting from the spherical $\nu {h}_{11/2}$ orbital, drive the deformation. Meanwhile, the extruder $\nu 9/{2}^{+}[404]$ orbital stabilizes the deformation at a saturation level of about $\beta \approx 0.4$. On the other hand, for the protons, the downsloping $\pi 1/{2}^{+}[440]$ and $\pi 3/{2}^{+}[431]$ orbitals, originating from the spherical $\pi {g}_{9/2}$ orbital, are fully occupied at Z = 38 and Z = 40, again at a deformation $\beta \approx 0.4$. These proton intruder orbitals have a large spatial overlap with the neutron intruder orbitals, creating a minimum in the binding energy at $\beta \approx 0.4$. At the neutron number N = 60 this deformed configuration is favored over the spherical one and deformation sets in rapidly.

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Clément et al [42, 99] measured the spectroscopic quadrupole moments of low-lying states in ⁹⁶Sr and ⁹⁸Sr and reduced transition probabilities between bands, using CE. The ⁹⁶Sr beam was produced by injecting CF₄ gas into the target ion source and extracting ⁹⁶Sr¹⁹F molecules. The otherwise dominant ⁹⁶Rb contamination was removed by setting the mass separator to the molecular mass at A = 115. The molecules were subsequently broken up in the charge breeder. In the case of ⁹⁸Sr, with a half-life of 653 ms, the ⁹⁸Rb (${T}_{\tfrac{1}{2}}=102$ ms) contaminant was suppressed by keeping the A = 98 ions sufficiently long in the REXTRAP for most of the Rb to decay to Sr, resulting in an 80% pure ⁹⁸Sr beam. The measurements of the spectroscopic quadrupole moments of excited ${2}^{+}$ states established shape coexistence of highly deformed prolate and spherical configurations in ⁹⁸Sr. A comparison of $B(E2)$ values and the spectroscopic quadrupole moments of the ${2}_{1}^{+}$ state in ⁹⁶Sr and the ${2}_{2}^{+}$ state in ⁹⁸Sr underlined their similarity and establishes the shape inversion at N = 60. In this work the interpretation of measured reduced E2 matrix elements using a phenomenological two-band mixing model substantiates the weak mixing between prolate and spherical configurations in the wave functions of the ${0}^{+}$ states in ⁹⁸Sr, in spite of their proximity in energy.

Albers et al [38, 39] measured the transition E2 matrix element between the ground and the first ${2}^{+}$ state, and the diagonal matrix element of the ${2}^{+}$ state, for ${}^{\mathrm{92,94,96}}$Kr, using CE. While the quadrupole moment increases significantly between N = 56 and N = 60, no sudden onset of deformation was observed.

Sotty et al [40] reported Coulomb-excitation measurements on the neutron-rich isotopes ⁹⁷Rb₆₀ and ⁹⁹Rb₆₂ produced as radioactive beams. The challenge for these experiments was the short beam half-lives, 169 ms and 54 ms respectively for ⁹⁷Rb and ⁹⁹Rb, that required short trapping and breeding times in REXTRAP and REXEBIS in order to minimize in-flight decay. The excited states in these odd-A nuclei, that form rotational bands built on the ground state, were observed for the first time. The results provide clear-cut evidence for enhanced quadrupole collectivity of the ${}^{\mathrm{97,99}}$Rb nuclei and firmly identify the deformation-driving configuration of the odd proton. Detailed information on the M1 transition strengths in ⁹⁷Rb provided concrete experimental evidence for the $\pi {g}_{9/2}[431]3{/2}^{+}$ Nilsson-configuration assignment.

CE experiments to study electromagnetic properties of radioactive even–even Hg isotopes were performed with 2.85 MeV/nucleon mercury beams from REX-ISOLDE. These represented the first time that such heavy beams were accelerated using the facility, paving the way for other studies of nuclei in this mass region and heavy nuclei in the octupole actinide mass region. For these heavy nuclei internal conversion plays an important role in the analysis and information from the K x-ray yield, enhanced over that expected from atomic processes arising from beam-target collisions [100], and β-decay studies (see [46] and references therein) were used to determine the E0 component of the ${2}_{2}^{+}$ to ${2}_{1}^{+}$ transition as well as the population of the ${0}_{2}^{+}$ state. In addition, independent lifetime measurements were made of excited states in ${}^{\mathrm{182,184,186}}$Hg (see [46] and references therein). From the combined data set, magnitudes and relative signs of the reduced E2 matrix elements that couple the ground state and low-lying excited states in ${}^{182\mbox{--}188}$Hg were extracted. Information on the deformation of the ground and the first excited ${0}^{+}$ states was deduced using the quadrupole sum rules approach, see figure 8. The results showed [46] that the ground state is slightly deformed and of oblate nature, while a larger deformation for the excited ${0}^{+}$ state was noted in ${}^{\mathrm{182,184}}$Hg. In contrast to that observed in the Sr, Zr isotopes with N = 60, the co-existing bands in the light mercury isotopes were shown to strongly mix. This accounts for the almost constant excitation energy observed for the first ${2}^{+}$ state in ${}^{182\mbox{--}188}$Hg.

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Coulomb excitation studies of ${}^{\mathrm{196,198},\mathrm{200,202}}$Po [47] and ${}^{\mathrm{202,204}}$Rn [48] were also carried out. For the polonium isotopes the observed behavior of the deformation of the ground state is consistent with charge radii measurements [101] that indicate the onset of significant deformation occurring for $A\lt 199$. Although there is a paucity of data for the ${0}_{2}^{+}$ and ${2}_{2}^{+}$ states in these nuclei, calculations using a two-level mixing model for the polonium isotopes hint at the spin-independent mixing of a more spherical structure with a weakly deformed oblate structure. For the radon isotopes matrix elements were determined for the ${2}_{1}^{+}\to {0}_{1}^{+}$ transitions and (for A = 202 only) the ${4}_{1}^{+}\to {2}_{1}^{+}$ and ${2}_{2}^{+}\to {2}_{1}^{+}$ transitions. The precision from this experiment is not sufficient to distinguish between oblate-, prolate-, and spherical-like charge distributions.