Pairing-quadrupole interplay in the neutron-deficient tin nuclei: first lifetime measurements of low-lying states in $^{106,108}$Sn

The lifetimes of the low-lying excited states $2^+$ and $4^+$ have been directly measured in the neutron-deficient $^{106,108}$Sn isotopes. The nuclei were populated via a deep-inelastic reaction and the lifetime measurement was performed employing a differential plunger device. The emitted $\gamma$ rays were detected by the AGATA array, while the reaction products were uniquely identified by the VAMOS++ magnetic spectrometer. Large-Scale Shell-Model calculations with realistic forces indicate that, independently of the pairing content of the interaction, the quadrupole force is dominant in the $B(E2; 2_1^+ \to 0_{g.s.}^+)$ values and it describes well the experimental pattern for $^{104-114}$Sn; the $B(E2; 4_1^+ \to 2_1^+)$ values, measured here for the first time, depend critically on a delicate pairing-quadrupole balance, disclosed by the very precise results in $^{108}$Sn. This result provides insight in the hitherto unexplained $B(E2; 4_1^+ \to 2_1^+)/B(E2; 2_1^+ \to 0_{g.s.}^+)<1$ anomaly.

a r t i c l e i n f o a b s t r a c t

Introduction
A little over a decade ago, the Sn isotopes were considered the paradigms of pairing dominance: low-lying states of good seniority, nearly constant J π = 2 + 1 excitation energies and parabolic B(E2; 2 + 1 → 0 + g.s. ) behavior. The latter was observed for A ≥ 116.

s. ) values in the light
Sn nuclei remains puzzling. The experimental and theoretical results that will be presented in this Letter provide a further insight, which reveals the underlying structure of the light tin isotopes, namely the counterbalance of quadrupole and pairing forces in the Sn isotopic chain.

Experiment
A multi-nucleon transfer reaction, that is commonly used to investigate neutron-rich nuclei [13][14][15], was unconventionally adopted to populate the Sn isotopes close to the proton drip line, so the isotopes of interest were populated in the collision of a 106 Cd beam and a 92 Mo target. The beam-target combination and beam energy were selected as a compromise between two requirements. On one hand the reaction fragments energy had to be sufficiently high to allow their identification by the spectrometer. On the other, in order to perform the lifetime measurement, the population of the states above the 6 + isomers had to be minimized; this condition imposed an upper limit on the excitation energy and consequently on beam energy, even at the expense of the cross section to populate more exotic species. The 106 Cd beam, provided by the separated-sector cyclotron of the GANIL facility at an energy of 770 MeV, impinged onto a 0.8 mg/cm 2 92 Mo target. The lifetime measurement was performed with the Recoil Distance Doppler-Shift (RDDS) method [16][17][18]. The target was mounted on the differential Cologne plunger with a 1.6 mg/cm 2 thick nat Mg degrader down stream. In order to measure the lifetimes of interest, 8 different target-degrader distances in the range 31-521 μm were used. The complete (A,Z) identification, together with the velocity vector for the projectile-like products was obtained on an eventby-event basis using the VAMOS++ spectrometer [19][20][21], placed at the grazing angle θ lab =25 • . In coincidence with the magnetic spectrometer, the γ rays were detected by the γ -ray tracking detector array AGATA [22,23], consisting of 8 triple-cluster detectors placed at backward angles in a compact configuration (18.5 cm from the target). The combination of the pulse-shape analysis [24] and the Orsay Forward-Tracking (OFT) algorithm [25] allowed to reconstruct the trajectory of the γ rays emitted by the fragments.
More details about the ion identification and the analysis procedure can be found in Refs. [26,27].

Results
Thanks to the precise determination of the ion velocity vector and the identification of the first interaction point of each γ ray inside AGATA, Doppler correction was applied on an eventby-event basis. The magnetic spectrometer directly measured the fragments velocity after the degrader (β af ter ≈ 9%). However, for each γ -ray transition two peaks were observed, related to its emission before and after the Mg foil: the γ rays emitted after the degrader are properly Doppler corrected, while those emitted before are shifted to lower energies because of the different velocity of the reaction fragment (β be f ore ≈ 10%). The relative intensities of the peaks as a function of the target-degrader distance are related to the lifetime of the state of interest. The areas are obtained via a χ 2 -minimization fit, performed by considering a gaussian shape of the peaks, whose centroid and FWHM were constrained from the closest and longest target-degrader distance spectra, and a linear background.
The complete identification of the VAMOS++ spectrometer and the employment of a multi-nucleon transfer (MNT) mechanism allowed us to reconstruct the Total Kinetic-Energy Loss (TKEL) of the reaction. This quantity was crucial for the lifetime measurements in 108 Sn, since it allows us to control the direct population of the excited states [18,28], in order to reduce possible contamination from the high-lying states above the 6 + 1 isomer. For this nucleus, in fact, the energies of the 2 + transitions are marked, indicating the unshifted (u) and shifted (s) centroids with a solid and a dashed line, respectively. In the inset, the TKEL distribution and the set gate (green) are presented. above the 6 + 1 isomer: for TKEL< 21 MeV, the 8 + 1 → 6 + 1 transition peaks became negligible and the measured lifetime of both 4 + 1 and 2 + 1 states remained constant, even for more restrictive conditions. The determination of such a TKEL condition and its effects on the lifetime measurement are discussed in detail in Ref. [29]. In that work, the ratio of the 4 + 1 → 2 + 1 transition components remains constant also for larger values of the TKEL, which is consistent with the unaltered line-shape of the Fig. 1 spectra at ≈ 900 keV; this is due to the fact that the 6 + 1 isomer "blocks" the depopulation from higher-lying states, so their presence does not affect the lifetime measurement of the 4 + 1 state. The same is obviously true for the 2 + 1 state but, in this specific case, a blind integration of the spectrum will include the unshifted component of the 8 + → 6 + transition. Being the 8 + a long lived state with a lifetime more than two orders of magnitude larger than the 2 + 1 , this would result in an artificially shorter lifetime. For this reason, the TKEL cut allows to get rid of the contamination of our transition of interest from the 8 + deexcitation. Fig. 2 (left) shows the Doppler-corrected γ -ray energy spectra of 108 Sn for several distances, requiring the TKEL< 21 MeV condition. This nontraditional procedure allowed us to take into account just the 6 + 1 , 4 + 1 and 2 + 1 states in the measurement of the lifetimes via Decay-Curve Method (DCM). Due to the direct population of the excited states, the presence of the longlived 6 + 1 isomer simplifies the lifetime measurement by "blocking" the depopulation from the higher-lying states and, as shown in Fig. 2 (right), it contributes just as an offset to the decay curves of the 4 + 1 and 2 + 1 states. For 106 Sn the described TKEL-gate procedure was not required and, because of the presence of the long-lived isomer, the decay cascade of the 6 + 1 , 4 + 1 and 2 + 1 states was taken into account while measuring the lifetime via DCM. Also in this case, the presence of the 6 + 1 isomeric state affects the lifetime analysis by introducing an offset, which is related to the direct population of the states [17], in the decay curves of the 2 + 1 and 4 + 1 excited states. The direct population of the excited states was extracted from the single-γ spectra. This information was used to constrain the parameters of the decay curves. The possible presence of additional feeders was investigated in the single-γ spectra and also in the γ -γ coincidences: except for those considered in the measurement, no other transitions feeding the 4 + 1 and 2 + 1 excited states were observed for both 106,108 Sn. In Fig. 2 (right pad) the decay curves are presented Table 1 Measured lifetime of the excited states I π in 106,108 Sn and corresponding values. The last column shows the theoretical predictions from the extension of the calculations of Ref. [1] (see text). for the 4 + 1 and 2 + 1 states: the extracted lifetime of the 2 + 1 state is in perfect agreement with the literature, supporting the validity of the experimental method. Therefore, thanks to the powerful setup and the unconventional experimental technique, the lifetime of the 2 + 1 and 4 + 1 states have been measured, for the first time, in 106,108 Sn. Table 1  The extracted B(E2) values for the 106,108 Sn isotopes are shown in Fig. 3  value for this isotope.

Discussion
In view of the present experimental results, the interpretation of the data in the neutron-deficient tin isotopes were performed within the theoretical companion contribution by Zuker [30]. This theoretical work explores the nuclear Hamiltonian within shellmodel calculations in the light Cd and Sn isotopes, considering the The context of these theoretical results is provided by the Pseudo-SU(3) symmetry, which acts in the space of gds orbits above (and except) g 9/2 . The basic idea is inspired by Elliott's SU(3) scheme [31,32] and consists in building intrinsic states that maximize the quadrupole operator [33][34][35]. As shown in Figure 2 of Ref. [30] (top right pad), for the light Sn nuclei only the first 6 neutrons play a role in the value of the quadrupole operator, while the contribution of the following 6 neutrons is null, leading to a "plateau" in the B(E2; 2 + Sn the quadrupole strength is much reduced due to the absence of g 9/2 proton holes. It is still strong enough to produce a stable B(E2; 2 + 1 → 0 + g.s. ) pattern analogous to the Cd one, but the B(E2; 4 + 1 → 2 + 1 ) behavior becomes sensitive to pairing and singleparticle behavior. Thus, it appears that the quadrupole dominance in Cd gives way to a form of pairing-quadrupole interplay in Sn. For the sake of completeness, in what follows we briefly explain the steps involved in the shell model calculations. The interaction must be extracted from a realistic potential, properly renormalized and monopole corrected. All realistic potentials give very similar results [34], N3LO was chosen [46] and V low−k -regularized [47]. The CDB [48] or AV18 [49] potentials would yield the same results. The indispensable renormalizations amount to a -rigorously established-30% boost of the quadrupole force and a phenomenological 40% increase of the pairing force [50]. The replacement of the monopole term is imperative, since the realistic interactions have bad monopole behavior [34, Sec. II.B.3]. Thus, this correction was done by replacing the monopole part of the interaction with the Hamiltonian provided by the GEMO (GEneral MOnopole) code [51], which is based on the Duflo-Zuker mass formula [52,53], adding the single particle spectrum of 101 Sn (in parentheses the energies in MeV):  is an extrapolated estimate (EX) equivalent to pushing up the s 1/2 orbital to 1.6 MeV, which results in the I.3.4s1.6 interaction.
The calculations were performed with the ANTOINE program [34] in ut M spaces in the gds shell of up to u g 9/2 -proton holes and t g 9/2 -neutron holes, for a total of M holes. Here we present ut M = 202 cases of m-scheme dimensions of up to 6 · 10 7 , but it was checked that they reproduce well the 10 10 -dimensional ut M = 444 cases. in the position of the s 1/2 orbit has an influence, albeit minor. In the Fig. 4 (b) both the pairing and the single particle shift make an enormous difference in the B(E2; 4 + 1 → 2 + 1 ) behavior. In the case of strong quadrupole dominance, the B(E2; 2 + tern would be as in Fig. 4 (a) and the B(E2; 4 + 1 → 2 + 1 ) one would be the same multiplied by 1.43. This is not far from the 1.25 for the I.3.0 case, as shown in Fig. 4 (c). The B 4/2 ratio is further reduced for I.3.2, while the proportionality between patterns is completely to account for the omission of the h 11/2 shell, that plays a small but significant role. Experimental B(E2; 2 + 1 → 0 + g.s. ) values are the weighted averages of the Fig. 3 results, while the B(E2; 4 + 1 → 2 + 1 ) data comes from Ref. [9] and the present work.
I.3.0s1.6 has the s 1/2 single particle energy moved up by 800 keV with respect to GEMO.
lost for the two I.3.4 cases, but both are close to the observed values in 112−114 Sn [9]. Moreover, our new measure in 108 Sn breaks the ambiguity in favor of the I.3.4 chosen standard with GEMO spectrum, providing a potentially interesting suggestion about the spectrum of 101 Sn. Traditionally all emphasis has been put in explaining the B(E2; 2 +  [12], whose B(E2; 2 + tern is identical to ours, while the wave-functions exhibit strong spin and mass dependence in the g 9/2 proton-hole occupancy, which are nearly constant in our case. We attribute the coincidence in the patterns to good monopole behavior. In addition to the B(E2) strengths, the theoretical estimations are in good agreement with the excitation energy of the 2 + 1 and 4 + 1 states: besides the I.3.0 that represents the no-pairing limit, the various interactions presented in the paper are within ≈ 200 keV accuracy.

Conclusions
The unconventional use of multi-nucleon transfer reactions with a plunger device has allowed us to measure lifetimes in the neutron-deficient 106,108 Sn isotopes. Since deep-inelastic reactions directly populate the low-lying excited states of the channels of interest, the experimental limitations caused by the presence of the low-lying isomers were overcome. Moreover, the TKEL-gate technique allowed us to avoid the possible contamination from highlying states above the 6 + 1 isomers. These results represent a step forward to our current knowledge of the region. Indeed, the precise measurement of the B(E2; 4 + 1 → 2 + 1 ) value in 108 Sn has led to a unique theoretical work which, in addition to discuss in detail the balance between pairing-quadrupole correlations in the nuclear interaction, questions the "goodness" of previous theoretical works. In fact, while many different theoretical works have claimed to reproduce the trend of the reduced transition probabilities B(E2; 2 + 1 → 0 + g.s. ) in the N = Z = 50 region, our paper highlights that the B(E2; 2 + 1 → 0 + g.s. ) values are not the only ones to be of key importance for understanding the nuclear structure close to 100 Sn; actually, the B(E2; 4 + 1 → 2 + 1 ) values, whose theoretical discussion is faced for the very first time in this article, result to be of crucial importance for understanding the nuclear structure of the region.
Further experimental and theoretical studies should focus their attentions on the electromagnetic properties not only of the 2 + transitions, but also of the 4 + 1 → 2 + 1 ones. This will allow us to shed light on the peculiar structure of the Sn isotopic chain, of the Z ≈ 50 region and of other regions where similar behavior of the B(E2) strengths have been observed, such as the N = 50 isotonic chain. In addition, the proton-and neutron-transfer spectroscopic factors will provide further information on the microscopic nature of the low-lying states in the Z ≈ 50 region.

Declaration of competing interest
The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.