Test of the πg7/2 subshell closure at Z=58

a Wright Nuclear Structure Laboratory, Yale University, New Haven, CT 06511, USA b Institute for Nuclear Research and Nuclear Energy, Bulgarian Academy of Sciences, 1784 Sofia, Bulgaria c Institut für Kernphysik, Universität zu Köln, 50937 Cologne, Germany d Argonne National Laboratory, Argonne, IL 60439, USA e Department of Chemistry and Biochemistry, University of Maryland, College Park, MD 20742, USA


Introduction
Studying collective excitations in nuclei gives insight into the mechanisms responsible for driving these strongly interacting many-body systems toward deformation. Highly correlated collective structures originate from a coherence in the independent motion of the neutrons and protons in a mean field modified by the residual interactions between the nucleons. Investigations of isoscalar and isovector excitations in a chain of isotopes provide extensive complementary information on the proton-neutron interaction. Often, the underlying single-particle structure has been found to influence the stability of such collective excitations, showing an interesting interplay between collective and the singleparticle degrees of freedom [1,2]. This competition results in an evolution of nuclear properties with N and Z , as well as with excitation energy and angular momentum. on collective mixed-symmetry states (MSSs) was observed. In the framework of the interacting boson model-2 [6,7], MSSs are described as excitations in which protons and neutrons move partially out of phase. Their fully-symmetric analog states (FSSs), i.e., 2 + 1 states in even-even nuclei, where the two types of nucleons move in phase, have similar configurations and are lower in excitation energy. A characteristic property of MSSs is their connection to FSSs with the same number of quadrupole bosons via strong M1 transitions.
In 138 Ce, the M1 transition strength between the higher-lying (2 + 1,ms ) mixed-symmetry level and the first excited 2 + state was found to be fragmented [5]. In contrast, in 134 Xe [3] and 136 Ba [4] the M1 strength remains largely concentrated in a single transition. Moreover, the total measured M1 strength is smaller for 138 Ce than for the other isotones. These observations were attributed to a lack of shell stabilization in 138 Ce [5], based on calculations within the quasiparticle-phonon-model (QPM) [8,9]. In this concept, the purity of the 2 + 1,ms state gets "washed out" in 138 Ce due to its single-particle structure. In a simplified independent-particle model, the complete filling of the π g 7/2 orbital at Z = 58 leads to configurations involving the higher-lying πd 5 [10], only the absolute B(M1; 2 + 4 → 2 + 1 ) strength has been measured [11], and no final conclusion can be drawn at this point. Large-scale Shell Model (LSSM) calculations were also carried out [12] to study the evolution of MSSs in the N = 80 isotones. The wave functions of low-lying states in all N = 80 isotones up to Z = 60 revealed significant mixing of the π g 7/2 and d 5/2 configurations and no pronounced shell closure was found [12]. Additional pairing strength was needed to achieve agreement in total M1 rates. However, the fragmentation of the M1 strengths was not reproduced quantitatively. Contrary to the QPM, the LSSM predicts an isolated MSS in 140 Nd.
Due to the conflicting conclusions on a π g 7/2 subshell closure within the two models, the main components of the states in question need to be verified experimentally. The magnetic moment of a state is a sensitive probe of its wave function. Therefore, a measurement of the g factor of the 2 + 1 level in Z = 58, 138 Ce was performed for the first time using Gammasphere and the Time Dependent Recoil Into Vacuum technique (TDRIV). The 2 + 1 level in this nucleus is the fully-symmetric analog of the 2 + 1,ms state [12]. Constraining the proton-neutron contributions in the wave function of the 2 + 1 state serves as a direct test for whether enhanced pairing strength is needed in the region, which impacts the structure and purity of MSSs. In addition, the simultaneous high- gives further insight into the existence of a possible subshell at Z = 58.

Experimental technique
Low-lying excited states in 138,142 Ce were populated via Coulomb excitation in inverse kinematics. 142 Ce and 138 Ce beams of intensity ∼1.7 enA and energies of 494 MeV and 480 MeV, respectively, were provided by the ATLAS accelerator at Argonne National Laboratory. The experimental setup consisted of the Yale plunger device [13] positioned at the center of the Gammasphere array [14] comprising 100 HPGe detectors arranged in 16 rings. The plunger hosted a 0.85-mg/cm 2 -thick 24 Mg target for Coulomb excitation, followed by a nat Cu stopper of 15.7 mg/cm 2 thickness that stops the beam but allows the target recoils to pass through. The target-to-stopper distance, d m , (relative to the point of electrical contact) was varied between ∼1 μm and 3 mm to enable a lifetime analysis of the states populated in the reaction with the Recoil Distance Doppler Shift (RDDS) method [15], as well as to measure the deorientation of the nuclear spin in vacuum. The Mg recoils were detected by a 300-mm-thick silicon detector kept at 0 • with respect to the beam axis, at a distance of ∼8 mm behind the target. This detector covered a laboratory solid angle of ±29.7 • . Angular distributions of deexciting γ rays were extracted at all target-to-stopper distances. A particle-γ coincidence, or a downscaled particle-singles or γ -singles event trigger, was required.
Transitions from excited states in 138,142 Ce are displayed in  and greater than 122 • with respect to the beam axis (see insets in Fig. 1). For angles near 90 • , Monte Carlo simulations were used to estimate the centroids and shapes of SH and US components. The Monte Carlo code [16,17] simulates the time behavior of the velocity of the ions of interest in three dimensions. It takes into account the reaction kinematics, the slowing down in the target and stopper, and the free flight in vacuum. Details about the determination of stopping powers can be found in Ref. [18]. A γ -ray spectrum for the 79.2 • detector ring is compared in Fig. 2 to the Monte Carlo simulations fitting the relative heights of the various peak components. A more detailed description of a similar analysis, including the fit of the stopping component at forward and backward angles is provided in Refs. [19,20].

Results
The angular distributions of the 2 + 1 → 0 + transitions are described by the standard formalism [21] for perturbed particle-γ correlations as where the Q k coefficients take into account the attenuation due to the finite solid angle of the Ge detectors [22], B k are the m state distribution coefficients, R k are the Racah coefficients [21], P k (cos(θ γ )) are the associated Legendre polynomials, and G k (t) are the time-dependent vacuum attenuation coefficients, first introduced on the basis of the theory of Abragam and Pound [23].
As the nuclei emerge from the target, a loss of alignment of the nuclear spin is observed due to hyperfine interactions. The resulting reduction in the anisotropy of the particle-γ angular correlations is represented by the G k (t) factor in Eq. (1). According to the static approach discussed in Refs. [19,[24][25][26][27][28], requiring the electronic lifetimes to be considerably larger than the average lifetime of the nuclear state, the deorientation coefficients G k (t) are related to the g factor of the state by Herein, the α k coefficients define the anisotropy of the γ angular distribution at t → ∞ [29] and C k are the hyperfine interaction parameters, which can be calibrated from a known g factor of a state in the same isotopic chain [24]. Using the TDRIV technique, relative g factors are obtained, and in this experiment, the known g(2 + 1 ) = 0.21 (5) Fig. 3. A relativistic Lorentzboost factor for the solid angle was introduced to correct the intensities of Doppler-shifted γ rays [31]. The degree of isotropy of the angular distributions increases with distance. This is a clear indication of deorientation due to the hyperfine interactions. To extract the G k (t) coefficients, the normalized experimental angular distribution curves were fitted to the function where  [19], the in-flight deorientation coefficients take into account the decay of the state and are given bỹ where λ is the decay constant. The deorientation coefficients for 138,142 Ce and for k = 2, 4, are displayed in Fig. 4.
To obtain the g factor of a nuclear state from the experimentally-deduced deorientation coefficients, the level lifetime, τ , is required. As a first estimate, the literature values for the 2 + 1 states in 138,142 Ce [30] were adopted. The resulting deorientation coefficients were fed back into a separate lifetime analysis described below in an iterative approach until convergence was achieved. Lifetimes were obtained via the RDDS method [15]. The ratios of the shifted intensities, I SH , corrected for the Lorentz boost and deorientation, to the total intensities, P = I SH /(I SH + I US ) are plotted as a function of distance in Fig. 5. A single exponential decay function of the form P (d) = 1 − exp(−λd/v) was fitted to the data to obtain the lifetime of the 2 + 1 level in 138 Ce. In the equation above, with d m being the measured distance between the target and the stopper, and d 0 is the offset, corresponding to the minimum achievable distance without electrical contact. Independent fits from 16 detector rings resulted in a precise determination of this offset, which was then fed into the fit of deorientation data. In 138 Ce, the population of higher-lying states is negligible (see Fig. 1). The 4 + 1 state at 1477 keV is populated with an approximate 1% probability with respect to the 2 + 1 level. The observed 4 + 1 → 2 + 1 transition has a dominant Doppler-shifted component, even for the smallest, 2 μm, target-to-stopper distance. Considering the velocity of the 138 Ce recoils, and the total flight distance including the 13 μm offset, a lifetime of less than 1 ps was estimated for the 4 + 1 state. The effect of this feeding on the intrinsic lifetime of the 2 + 1 state in 138 Ce is less than 1% and is included in the adopted uncertainty. In 142 Ce, a feeding correction from the higher-lying 4 + 1 level with a lifetime of 10.8(10) ps [30] had to be considered. The decay probability curve for this nucleus was fitted with a combined exponential decay function of the form Here, P 0 , λ 0 and P 1 , λ 1 are the respective population probabilities and decay constants of the 2 + 1 and 4 + 1 states. For 142 Ce, P 0 was 97% and P 1 3%, as given by the Winther-De Boer code [32]. These numbers were verified by the measured 4 + 1 → 2 + 1 and 2 + 1 → 0 + intensities. Lifetimes were obtained for the 16 detector rings and the weighted averages are adopted in Table 1 in comparison to literature values, along with the extracted B(E2)↓ values for 138,142 Ce. The present results are in good agreement with previous measurements and present a significant improvement in the

Discussion
Assuming a positive sign for the g(2 +  The new data discriminate between two LSSM predictions by the Strasbourg group [12] for 138 Ce. The first used the GCN5082 interaction, derived from a realistic nucleon-nucleon Bonn-C potential within the gdsh valence space, and the second was obtained after modifying the pairing matrix elements of the same interaction. The two calculations led to significantly different results for g factors. As seen in Fig. 6(b), the result from the original interaction is excluded by the measurement by more than 2σ , whereas the prediction with enhanced pairing lies within the present error bar. Hence, modifications in the pairing interaction, as applied to the GCN5082 interaction, appear to be necessary for a good description of the nuclear wave functions at Z = 58. In contrast, when considering also the 132 Te [24], 134 Xe [33] and 136 Ba [34] isotones, the recent data are markedly different from the predictions of the interaction with enhanced pairing. The original GCN5082 interaction appears to reproduce the observations better below Ce. This may indicate that an enhanced pairing interaction is of higher importance at Z = 58, where the π g 7/2 orbital is filled.
In recent work on 140 Nd [35], a modest suppression of the B(E2)↓ strength at Z = 58 had been discussed as a signature of the subshell closure at Z = 58. However, the literature B(E2)↓ value had an error bar of about 8%. In the present TDRIV experiment, level lifetimes are obtained with high precision, and the statistical error is reduced to about 1.5%. The deviation from a near-linear trend in B(E2)↓ values pointed out in Ref. [35] is now confirmed and the new B(E2)↓ value in 138 Ce agrees well with the LSSM prediction, similar to that observed for the lower-Z isotones.
To summarize, the g(2 + 1 ) factor in 138 Ce was measured relative to that in 142 Ce employing the TDRIV technique in inversekinematics Coulomb excitation. The resulting value is in good agreement with a LSSM calculation explicitly modifying pairing matrix elements. The measured B(E2)↓ value in 138 Ce supports a subshell closure at Z = 58. These results underline the conclusion from Ref. [12] that the pairing interaction is important for a correct understanding of proton-neutron symmetry in low-lying collective states at Z = 58. In order to differentiate between, and to give stronger constraints to QPM and LSSM calculations, the reference g (2 +