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

TL;DR

This study measures the lifetimes of low-lying states in neutron-deficient $^{106,108}$Sn isotopes, revealing the dominance of quadrupole forces and a delicate pairing-quadrupole balance affecting $B(E2)$ transition strengths.

Contribution

First lifetime measurements of low-lying states in $^{106,108}$Sn, providing new insights into pairing-quadrupole interplay in these nuclei.

Findings

Quadrupole force dominates $B(E2; 2_1^+ o 0_{g.s.}^+$) values.

Precise $B(E2; 4_1^+ o 2_1^+)$ measurements reveal pairing-quadrupole balance.

Results explain the $B(E2)$ anomaly in $^{108}$Sn.

Abstract

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.

Tables

Table 1. Table 1: Measured lifetime of the excited states I π superscript 𝐼 𝜋 I^{\pi} in 106,108 Sn and corresponding B ​ ( E ​ 2 ; I π → I π − 2 ) 𝐵 → 𝐸 2 superscript 𝐼 𝜋 superscript 𝐼 𝜋 2 B(E2;I^{\pi}\to I^{\pi}-2) values. The last column shows the theoretical predictions from the extension of the calculations of Ref.  Banu et al. ( 2005 ) (see text).

	$I^{\pi }$	Eγ [keV]	$\tau$ [ps]	B(E2) [$e^{2}f{m}^4$]
				Exp.	Theo.
108Sn	$2_1^{+}$	1206.1	0.76 (8)	422 (44)	425
	$4_1^{+}$	905.1	3.7 (2)	364 (20)	349
106Sn	$2_1^{+}$	1207.7	5 (4)	245 (132)	339
	$4_1^{+}$	811.9	1.3 (7)	446 (334)	379

Full text

A little over a decade ago, the Sn isotopes were considered the paradigms of pairing dominance: low-lying states of good seniority, nearly constant $J^{\pi}=2_{1}^{+}$ excitation energies and parabolic $B(E2;2_{1}^{+}\to 0_{g.s.}^{+})$ behavior. The latter was observed for $A\geq 116$. For the lighter species, experimental results on transition probabilities were scarce as the presence of low-lying isomeric states hindered direct measurements of lifetimes below them. From Coulomb-excitation measurements with radioactive ion beams only the reduced transition probability between the first excited $2^{+}$ state and the ground state could be determined Banu et al. (2005); Vaman et al. (2007); Cederkäll et al. (2007); Ekström et al. (2008); Kumar et al. (2010); Bader et al. (2013); Doornenbal et al. (2014); Kumar et al. (2017). Within experimental uncertainties, they suggest a rather-constant behavior for $106\leq A\leq 110$, instead of the parabolic trend expected when isovector $T=1$ pairing dominates. This “plateau” includes also the stable 112,114Sn nuclei, for which measurements of both $B(E2;2_{1}^{+}\rightarrow 0_{g.s.}^{+})$ and $B(E2;4_{1}^{+}\rightarrow 2_{1}^{+})$ values exist Jonsson et al. (1981). Before this work, the $B(E2;4_{1}^{+}\to 2_{1}^{+})$ values were completely absent in the neutron-deficient Sn isotopes.

The experiment described in this Letter was devoted to determine the strength of $2_{1}^{+}\to 0_{g.s.}^{+}$ and $4_{1}^{+}\to 2_{1}^{+}$ transitions in 106,108Sn by measuring the lifetime of $2_{1}^{+}$ and $4_{1}^{+}$ states. Although several theoretical interpretations have been proposed Ansari (2005); Banu et al. (2005); Ekström et al. (2008); Maheshwari et al. (2016); Togashi et al. (2018), the evolution of the $B(E2;2_{1}^{+}\to 0_{g.s.}^{+})$ 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.

A multi-nucleon transfer reaction, that is commonly used to investigate neutron-rich nuclei Szilner et al. (2007); Corradi et al. (2009); Valiente-Dobón (2016), 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 106Cd beam and a 92Mo 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 106Cd beam, provided by the separated-sector cyclotron of the GANIL facility at an energy of 770 MeV, impinged onto a 0.7 mg/cm2 92Mo target. The lifetime measurement was performed with the Recoil Distance Doppler-Shift (RDDS) method Dewald et al. (1989, 2012); Valiente-Dobón et al. (2009). The target was mounted on the differential Cologne plunger with a 1.6 mg/cm2 thick natMg degrader down stream. In order to measure the lifetimes of interest, 8 different target-degrader distances in the range 31-521 $\mu$m were used. The complete (A,Z) identification, together with the velocity vector for the projectile-like products was obtained on an event-by-event basis using the VAMOS++ spectrometer Pullanhiotan et al. (2008); Rejmund et al. (2011); Vandebrouck et al. (2016), placed at the grazing angle $\theta_{lab}$=25∘. In coincidence with the magnetic spectrometer, the $\gamma$ rays were detected by the $\gamma$-ray tracking detector array AGATA Akkoyun et al. (2012); Clément et al. (2017), 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 Bruyneel et al. (2013) and the Orsay Forward-Tracking (OFT) algorithm Lopez-Martens et al. (2004) allowed to reconstruct the trajectory of the $\gamma$ rays emitted by the fragments. More details about the ion identification and the analysis procedure can be found in Refs. Siciliano et al. (2017); Siciliano (2017).

Thanks to the precise determination of the ion velocity vector and the identification of the first interaction point of each $\gamma$ ray inside AGATA, Doppler correction was applied on an event-by-event basis. The magnetic spectrometer directly measured the fragments velocity after the degrader ($\beta_{after}\approx 9\%$). However, for each $\gamma$-ray transition two peaks were observed, related to its emission before and after the Mg foil: the $\gamma$ 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 ($\beta_{before}\approx 10\%$). The relative intensities of the peaks area as a function of the target-degrader distance are related to the lifetime of the state of interest. Figure 1 (left pad) shows the Doppler-corrected $\gamma$-ray energy spectra of 108Sn for several distances, where a Total Kinetic Energy Loss (TKEL) gate has been imposed in order to reduce the direct feeding of the higher-lying states Valiente-Dobón et al. (2009); Mengoni et al. (2009). For this nucleus the energies of $8_{1}^{+}\to 6_{1}^{+}$ and $2_{1}^{+}\to 0_{g.s.}$ transitions are similar, so traditional methods cannot be used to measure the lifetime of the $2_{1}^{+}$ state. Thus, the TKEL$\leq 21$ MeV condition was imposed until the $8_{1}^{+}\to 6_{1}^{+}$ transition peaks became negligible and the measured lifetime of both $4_{1}^{+}$ and $2_{1}^{+}$ states remained constant, even for more restrictive conditions; such a 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). In Fig. 1 (right pad) the decay curves are presented 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. For 106Sn 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 have been taken into account while measuring the lifetime via DCM. Therefore, thanks to the powerful setup and the unconventional experimental technique, the lifetime of the $2_{1}^{+}$ and $4_{1}^{+}$ states has been measured, for the first time, in 106,108Sn. Table 1 summarizes the experimental results, showing the lifetimes and the derived reduced transition probabilities $B(E2)$ for 108Sn and 106Sn isotopes, as well as the theoretical values from the extension of the calculations of Ref. Banu et al. (2005), obtained by employing the same interaction in the full $gds$ valence space for both protons and neutrons, using the effective charges $(e_{\pi},e_{\nu})=(1.35,0.65)$, allowing up to $4p-4h$ excitations and without any seniority truncation. The extracted $B(E2)$ values for the 106,108Sn isotopes are shown in Fig. 2 together with all previous experimental results for the whole isotopic chain. The $B(E2;2_{1}^{+}\rightarrow 0_{g.s.}^{+})$ strengths previously measured are compatible with the results obtained in this experiment, while for the $B(E2;4_{1}^{+}\rightarrow 2_{1}^{+})$ values no data existed in this region. Unfortunately, the exoticity of 106Sn and the necessity to avoid the population of the states above the long-lived $6^{+}$ isomer result in a rather large statistical error on the $B(E2;4_{1}^{+}\to 2_{1}^{+})$ value for this isotope.

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 Zuker (2019), this time taking into account also the newly measured $B(E2;4^{+}_{1}\rightarrow 2^{+}_{1})$ values. Given a proper interaction it is very simple to describe the “anomalous” $B(E2;2_{1}^{+}\to 0_{g.s.}^{+})$ pattern of the Sn isotopes while the—hitherto unexplored—$B(E2;4_{1}^{+}\to 2_{1}^{+})$ behavior demands more care.

The interaction must be extracted from a realistic potential, properly renormalized and monopole corrected. All realistic potentials give very similar results Caurier et al. (2005), N3LO was chosen Entem and Machleidt (2002) and $V_{\text{low-}k}$-regularized Bogner et al. (2003). The CDB Machleidt et al. (1996) or AV18 Wiringa et al. (1995) 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 Dufour and Zuker (1996). The monopole corrections were done by making the interaction monopole-free and adding the single particle spectrum of 101Sn given in GEMO Duflo and Zuker (1999) (in parenthesis energies in MeV):

$g_{9/2}(-6.0),\,d_{5/2}(0.0),\,g_{7/2}(0.5),\,s_{1/2}(0.8),\,d_{3/2}(1.6)$ .

The resulting interaction is called I.3.4. Possible uncertainties concern the pairing content, so we have also tried I.3.0 and I.3.2 (no and 20% corrections, respectively). Furthermore, according to the study of Ref. (Faestermann et al., 2013, Fig. 3.2.1), an alternative to the GEMO spectrum (DZ in that Figure) 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 Caurier et al. (2005) in $utM$ 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 $utM=202$ cases of m-scheme dimensions of up to $6\cdot 10^{7}$, but it was checked that they reproduce well the $10^{10}$-dimensional $utM=444$ cases.

Figure 3 (a) establishes that the amount of pairing makes no difference in the $B(E2;2_{1}^{+}\to 0_{g.s.}^{+})$ transition probabilities. The shift in the position of the $s_{1/2}$ orbit has an influence, albeit minor. This behavior may explain why the pattern in the calculations of Togashi Togashi et al. (2018) is identical to ours, while the wavefunctions are radically different Zuker (2019) in that they exhibit strong spin and mass dependence in the $g_{9/2}$ proton-hole occupancies, which are nearly constant in our case.

In the Fig. 3 (b) both the pairing and the single particle shift make an enormous difference in the $B(E2;4_{1}^{+}\to 2_{1}^{+})$ behavior.

The context of these results is provided by the Pseudo-SU(3) symmetry which acts in the space of $sdg$ orbits above (and except) $g_{9/2}$. For a strong enough quadrupole force and nearly degenerate single-particle spectrum, the system exhibits rotational features. In particular $B_{4/2}\equiv B(E2;4_{1}^{+}\to 2_{1}^{+})/B(E2;2_{1}^{+}\to 0_{g.s.}^{+})\approx 1.43$ (Alaga rule). All this is explained in Zuker (2019).

In the case of strong quadrupole dominance, the $B(E2;2_{1}^{+}\to 0_{g.s.}^{+})$ pattern would be as in Fig. 3 (a) and the $B(E2;4_{1}^{+}\to 2_{1}^{+})$ one would be the same multiplied by 1.43. This is close to the I.3.0 case, though the quotient (Fig. 3 (c)) is reduced to about 1.25. It is further reduced for I.3.2. In the two I.3.4 cases, the proportionality between patterns is completely lost, but both are close to the observed values in 112-114Sn Jonsson et al. (1981). Our new measure in 108Sn breaks the ambiguity in favor of the I.3.4 chosen standard with GEMO spectrum, providing a potentially interesting suggestion about the spectrum of 101Sn.

The $B_{4/2}<1$ anomaly had been observed in 114Xe de Angelis et al. (2002), in 114Te Möller et al. (2005) and, more recently, in 172Pt Cederwall et al. (2018), where it is stressed that no theoretical explanation is available, except through pairing dominance. The present calculations bring a change in that they account for observation and point to the underlying mechanism: it is clear that quadrupole dominance holds for the $J=0_{1},\,2_{1}$ states, but is strongly challenged for the $4_{1}$ state through mixing with a pairing dominated intruder. What is missing is a characterization of such an intruder. A novel situation that demands further study.

In summary, the unconventional use of multi-nucleon transfer reactions with a plunger device has allowed to measure lifetimes in the neutron-deficient 106,108Sn isotopes. The TKEL-gate technique was used to overcome the experimental limitations due to the presence of low-lying isomers, leading to the first $B(E2;4_{1}^{+}\to 2_{1}^{+})$ measurement in this mass region. The theoretical results show that the experimental trend of the $B(E2;2_{1}^{+}\to 0_{g.s.}^{+})$ values in the mass region $104\leq A\leq 114$ can be reproduced within the $gds$ model space. Traditionally, pairing was thought to be dominant in the Sn isotopes but the calculations indicate that the rather constant pattern of the $B(E2;2_{1}^{+}\to 0_{g.s.}^{+})$ values is associated to quadrupole dominance, independent of the pairing strength. However, in the case of $B(E2;4_{1}^{+}\to 2_{1}^{+})$ values pairing takes its revenge by becoming crucial. What is beyond doubt is the importance of the very precise measurements in 108Sn, which have shown to open new perspectives in the understanding of the quadrupole-pairing interplay.

The authors would like to thank the AGATA and VAMOS collaborations. Special thanks go to the GANIL technical staff for their help in setting up the apparatuses and the good quality beam and to A. Poves for his fruitful comments. This work was partially supported by the European Union’s Seventh Framework Programme for Research and Technological Development (grant no. 262010). The work was also supported (J.N.) by Swedish Research Council, (B.S.) by the Scientific and Technological Council of Turkey (TUBITAK) under the project no. 114F473, (A.G., R.P. and A.J.) by the Ministerio de Economía y Competitividad and the Ministry of Science of Spain under the grant agreements nos. EEBB-I-15-09671, FPA2017-84756-C4 and SEV-2014-0398 and by the EU-FEDER funds, (T.M and S.S.) by the Croatian Science Foundation under the project no. 7194, (I.K and D.S.) by the Hungarian National Research and Innovation Office (NKFIH) under the project nos. K128947, PD124717 and GINOP-2.3.3-15-2016-00034, (M.P.) by the Polish National Science Centre with the grants nos. 2014/14/M/ST2/00738, 2016/22/M/ST2/00269 and 2017/25/B/ST2/01569 under the COPIN-IN2P3, COPIGAL and POLITA projects.