Interplay of quasiparticle and vibrational excitations: First observation of isomeric states in 168Dy and 169Dy

Interplay of quasiparticle and vibrational excitations: First observation of isomeric states in 168Dy and 169Dy G.X. Zhang a, H. Watanabe a,b,∗, G.D. Dracoulis c,1, F.G. Kondev d, G.J. Lane c, P.H. Regan e,f, P.-A. Söderström b, P.M. Walker e, K. Yoshida g, H. Kanaoka h, Z. Korkulu i, P.S. Lee j, J.J. Liu k, S. Nishimura b, J. Wu b,l, A. Yagi h, D.S. Ahn b, T. Alharbi m, H. Baba b, F. Browne n, A.M. Bruce n, M.P. Carpenter d, R.J. Carroll e, K.Y. Chae o, C.J. Chiara p,2, Zs. Dombradi i, P. Doornenbal b, A. Estrade q, N. Fukuda b, C. Griffin q, E. Ideguchi r, N. Inabe b, T. Isobe b, S. Kanaya h, I. Kojouharov s, T. Kubo b, S. Kubono b, N. Kurz s, I. Kuti i, S. Lalkovski e, T. Lauritsen d, C.S. Lee j, E.J. Lee o, C.J. Lister d, G. Lorusso b,e,f, G. Lotay e, E.A. McCutchan d, C.-B. Moon t, I. Nishizuka u, C.R. Nita n,v, A. Odahara h, Z. Patel e, V.H. Phong b,w, Zs. Podolyák e, O.J. Roberts x, H. Sakurai b, H. Schaffner s, D. Seweryniak d, C.M. Shand e, Y. Shimizu b, T. Sumikama u, H. Suzuki b, H. Takeda b, S. Terashima a, Zs. Vajta i, J.J. Valiente-Dóbon y, Z.Y. Xu k, S. Zhu d


Available online 22 October 2019
Editor: D.F. Geesaman excitation energy of 1378 keV, and its presence affirmed independently using γ -γ -γ coincidence data taken with Gammasphere via two-proton transfer from an enriched 170 Er target performed at Argonne National Laboratory. This isomer is assigned J π = K π = (4 − ) based on the measured transition strengths, decay patterns, and the energy systematics for two-quasiparticle states in N = 102 isotones.
The underlying mechanism of two-quasiparticle excitations in the doubly midshell region is discussed in comparison with the deformed QRPA and multi-quasiparticle calculations. In 169 Dy, the B(E2) value for the transition de-exciting the previously unreported K π = (1/2 − ) isomeric state at 166 keV to the K π = (5/2 − ) ground state is approximately two orders of magnitude larger than the E2 strength for the corresponding isomeric-decay transition in the N = 103 isotone 173  One of the fundamental and universal features of nuclear systematics observed across the chart of nuclides is that the nucleus incurs a deviation from the spherical equilibrium shape when increasing the proton/neutron valency moving away from the magic numbers. Rare-earth (RE) isotopes in the mass range from 150 to 180 are known to be well deformed even in their low-lying states. Despite the predicted smooth variation of axial deformation as a function of the proton number ( Z ) or neutron number (N) in the doubly midshell region [1][2][3][4], there are some irregularities in the systematics of the observed first 2 + energies for even-even RE nuclides: local minima emerge at N = 98 for 62 Sm, 64 Gd, 66 Dy and at N = 104 for Dy, 68 Er, 70 Yb isotopes (see Fig. 30 in Ref. [5]). This anomalous behavior of the 2 + energy has been discussed in terms of deformed sub-shell closures [6,7], which can stabilize the nuclear deformation and thereby have a significant impact on the formation of the so-called RE element peak around A = 165 in the r-process solar abundance distribution [8,9]. The presence of energy gaps in the deformed single-particle space should also influence the excitation spectra in well-deformed nuclei, which are characterized by rotational and vibrational motion, as well as by other non-collective excitations like K isomers (K denotes the angular momentum projection on the symmetry axis of the deformed nucleus). The competition between such different intrinsic excitations makes the excited level structure more complicated than a simple coupling scheme predicts, due to possible configuration mixing. To provide a good testing ground for more advanced nuclear structure models that can be incorporated in any r-process network simulations, detailed spectroscopic studies of the excited states in this largely deformed region are highly demanded.
In this Letter, we present previously unreported isomeric states and their decay properties identified in 168 Dy 102 and 169 Dy 103 , which correspond to double-and single-hole systems relative to the valence maximum nucleus 170 Dy located at the double midshell Z = 66 and N = 104. The presence of isomers with halflives of the order of several tens of nanoseconds or longer often facilitates exploration of exotic nuclei produced in fragment separators by measuring delayed γ -rays that are coincident with the identified particles on an event-by-event basis [10], as demonstrated recently for a number of neutron-rich RE isotopes at the Radioactive-Isotope Beam Factory (RIBF) [11][12][13][14][15][16][17]. Besides the usefulness to populate excited levels, characteristic isomers can serve as an effective probe for the underlying nuclear structure, such as single-particle orbitals, pairing correlations, and collective motions [18]. Thus, the present work can provide a significant insight into the structure of midshell nuclides that were hard to reach previously.
We have analyzed experimental data obtained from two independent experiments; one using the EURICA spectrometer at RIBF [19], and the other using the Gammasphere array at Argonne Tandem-Linac Accelerator System (ATLAS) facility [20]. At RIBF, neutron-rich nuclides around A = 170 were produced by in-flight fission of a 238 U 86+ beam at 345 MeV/u with an average intensity of 12 pnA, incident on a 5-mm thick Be target. The nuclei of interest were separated and identified through the in-flight separator BigRIPS [21]. Identification of particles with the atomic number ( Z ) and the mass-to-charge ratio ( A/Q ) was achieved on the basis of the E-TOF-Bρ method, in which the energy loss ( E), time of flight (TOF), and magnetic rigidity (Bρ) were measured using the focal-plane detectors on the beam line. In total about 1500, 9000, 3600, and 1700 ions were collected for 168 Dy 66+ , 168 Dy 65+ , 169 Dy 66+ , and 169 Dy 65+ , respectively, all of which were involved in the present data analysis. The identified particles were implanted into WAS3ABi [19], which consisted of two double-sided silicon-strip detectors (DSSSD) stacked compactly. Each DSSSD had a thickness of 1 mm with an active area segmented into 60 and 40 strips (1-mm pitch) on each side in the horizontal and vertical dimensions, respectively. The DSSSDs also served as detectors for electrons following β-decay and internal conversion processes [22].
Gamma rays emitted following the heavy-ion implantation and their subsequent radioactive decay were detected by the EURICA γ -ray spectrometer [19,23], consisting of 12 Cluster-type detectors, each of which contained seven HPGe crystals packed closely. More detailed information on the experimental setup and data analysis can be found in Ref. [5].
In the experiment at ATLAS, excited states in 168 Dy were populated via two-proton removal from an 170 Er target, bombarded by a 136 Xe beam at 830 MeV. The target was composed of a 6 mg/cm 2 metallic foil, enriched in 170 Er, and placed on a gold backing with a thickness of 25 mg/cm 2 , thick enough to stop the reaction products at the target position. Under these conditions, the effective beam energy ranges from about 20% above the Coulomb barrier at the entrance to near the barrier at the rear of the target. Gamma rays were detected with the Gammasphere array [20]. More detailed descriptions of the experimental conditions are presented in Ref. [24]. Gamma-ray coincidence measurements with a requirement that three or more Ge detectors fired, in conjunction with pulsed beams separated by 825 ns, complemented the results obtained at RIBF for isomers with half-lives in the nanosecond to microsecond range.
The level schemes of 168 Dy and 169 Dy established in the present work are displayed in Fig. 1, where the levels assigned different K π quantum numbers are described in different colors. The reader is referred to the online version of this article.
Prior to the present work, the excited states of 168 Dy had been studied by the β decay of 168 Tb [27] and multi-nucleon transfer reactions with a 82 Se beam incident on an 170 Er target [28]. The γ rays at energies of 75 and 173 keV, which were previously assigned as (2 + 1 ) → 0 + 1 and (4 + 1 ) → (2 + 1 ), respectively, have been confirmed in delayed coincidence with 168 Dy ions in the EURICA experiment, as shown in Figs. 2(a) and 2(b). In this analysis for isomeric-decay transitions, however, the 227-keV γ ray reported in Ref. [27] and the yrast E2 sequence above the (4 + 1 ) state of the ground-state rotational band identified in Ref. [28] have not been observed, due presumably to the difference in feeding pattern. Meanwhile, two previously unreported γ rays at 915 and 236 keV can be unambiguously seen in Fig. 2 In the EURICA experiment, γ rays following the β decay of 168 Tb have also been observed with higher statistics than in Ref. [27]. The results of the β-γ analysis will be presented elsewhere [26]. Here, for the sake of identification of the isomeric state in 168 Dy, we use only the fact that the 915-keV γ ray is coincident with the 75-keV γ ray, but not with the 173-keV one [see Fig. 2(d)], thus being assigned as feeding the (2 + 1 ) state. An apparent β feeding to the 990-keV state from the parent nucleus 168 Tb, which is expected to have a ground-state spin and parity of (4 − ) with the π3/2 + [411] ⊗ ν5/2 − [512] configuration [27], allows for a tentative assignment of J π = (3 − ) [26]. Note that the (4 − ) assignment for 168 Tb is based on the consideration of the spins and parities, as well as the corresponding Nilsson orbits, assigned to the respective ground states of the neighboring odd-A Tb isotopes and N = 103 isotones [27]. The negative-parity level in 168 Dy is a candidate for the J π = 3 − member of the K π = 2 − octupole-vibrational band, which is known to emerge at compara- On the basis of the arguments presented above, a new isomeric state is identified at an excitation energy of 1378 keV in 168 Dy. A half-life of 0.57(7) μs has been derived from a log-likelihood fit of the summed γ -ray gated time spectra for the isomeric-decay transitions, as shown in the inset of Fig. 2(a). The efficiency-corrected γ -ray intensities (I γ ) relative to the 236-keV line are summarized in Table 1. It should be noted that, despite the observation of the 173-keV γ ray, any other γ rays which feed the (4 + 1 ) state can not be found in Fig. 2(a). If only one transition (e.g. an 1130-keV γ ray from the isomer) populated the (4 + 1 ) state, the total transi- Table 1 Summary of transitions from the K π = (4 − ) isomer in 168 Dy. (13) 38 (14) 1.
a Relative to the γ -ray intensity of the 236-keV transition tion intensity (I tot ) would be balanced with the following 173-keV transition, yielding a peak area with ∼ 10 counts at the corresponding energy, enough to be observed in Fig. 2(a). Therefore, the non-observation of such connecting γ rays implies that the (4 +
As will be discussed later, the K π = (4 − ) and (  The 1227-keV state prefers to decay towards the 990-keV state rather than the ground-state rotational band at lower excitation energy, implying that the 236-keV transition is of a fast E1 character. Such kind of "interband" E1 transitions have been known to take place between the K π = 2 + (γ ) and K π = 2 − (octupole) bands in the lighter even-A Dy isotopes [29]. Therefore, the 1227-keV state is expected to have either J π = 3 + or 4 + with K π = 2 + , which is populated via the 152-keV, E1 transition from the J π = K π = (4 − ) isomeric state. The E1 hindrance F W is about two orders of magnitude larger for the 152-keV branch than for the 40-keV transition, the other isomeric-decay branch towards the K π = (3 + ) state at 1338 keV (see Table 1). This is most likely ascribed to the difference in the K quantum number of the states to which the K π = (4 − ) isomer decays. Based on the consideration of the observed feeding pattern and hindrance, the 1227-keV state is associated with the K π = (2 + ) γ -vibrational band.
The energy systematics of the J π = K π = (3 + ), (4 − ), and (6 − ) states observed for the N = 102 isotones with Z = 62 − 72 are shown in the upper panel of Fig. 3. They appear at excitation energies approximately equal to the pairing gap, 2 ∼ 1.5 MeV, in this mass region, being characteristic of two-quasiparticle (2qp) excitations in even-even nuclei that involve Nilsson orbits π [Nn z ] near the Fermi surface. It should be noted that a K π = 6 − state has not been identified for 168 Dy in the present work. That level is expected to emerge at around 1.6 MeV from the energy systematics, and to preferentially decay towards the lower-lying K π = 4 − band with a short lifetime, as found for 170 Er (T 1/2 = 4 ns) [24], being presumably hard to observe with the present experimental sensitivity.
For 169 Dy, no spectroscopic information had been reported before the present work. Fig. 2(e) exhibits a γ -ray energy spectrum measured in coincidence with the identified 169 Dy ions within 0.35 − 6 μs. A single γ ray observed at 166 keV can be interpreted as an E2 transition arising from a J π = K π = (1/2 − ) isomer towards the presumed (5/2 − ) ground state [see Fig. 1  isotones are plotted in the upper and lower panels of Fig. 3, respectively.
It is to be noted that, although the aforementioned spin-parity assignments rely on the level systematics and plausible arguments on the observed transition strengths and decay patterns, all the individual arguments are convincing and ensure the full picture.
The excitation energies of the K π = (3 + ) states in the N = 102 isotones vary smoothly around 1.2 MeV with a minimum at Z = 70. To understand the underlying mechanism of the K π = 3 + excitation from a microscopic viewpoint, we have performed model calculations based on a Skyrme energy-density-functional (EDF) in the same framework as that applied to the K π = 2 + γ -vibrational excitation in our previous work [4,13]. With the use of the SkM* functional [35], the ground states are calculated by the Hartree  It is noteworthy that low-lying K π = 3 + bands are systematically identified in even-even nuclei with N = 98 − 104 and Z = 68 − 72 [29], and that the J π = 4 + members are strongly populated by deuteron and alpha inelastic scattering [37][38][39] that tends to favor collective excitations. The measured E4 strengths suggest the presence of hexadecapole vibrational nature [37][38][39].
In order to understand the K isomerism in the doubly midshell region, multi-quasiparticle calculations were performed for 168,170,172 Dy in a similar way to an approach in Ref. [40], but with the inclusion of particle-number conservation by means of the Lipkin-Nogami prescription for the pairing correlations [41]. As the model implements the so-called "blocking effect", the calculated pairing gaps and the Fermi level become configuration dependent. The energies and ordering of deformed single-particle (SP) levels were initially calculated based on the Woods-Saxon potential with universal parameters for fixed deformations β 2 , β 4 , and β 6 taken from Ref. [3], and then modified so as to reproduce approximately the empirical 1qp states in the neighboring odd-A nuclei. The left part of Fig. 5 compares the original and modified SP energies around the Z = 66 and N = 102 Fermi surfaces. The most striking difference between the two sets of SP levels is the inversion of the ν1/2 − [521] and ν7/2 + [633] orbits, which accounts for the appearance of the K π = 3 + state at low excitation energy in the N = 102 isotones, as discussed above. The rising of the ν1/2 − [521] level also causes a sizable gap at N = 98 between the ν5/2 − [523] and ν7/2 + [633] orbitals [7].
The results of the multi-quasiparticle calculations obtained with the modified SP levels are shown in the right part of Fig. 5.
Here, the proton and neutron 2qp states were calculated independently, and subsequently corrected for the empirical residual interactions that depend on the configurations involved, i.e., the spin-singlet coupling is somewhat lowered in energy, while the energetically unfavored spin-triplet coupling is raised, following the Gallagher-Moszkowski rule [42].  two configurations are expected to be strongly mixed since they are located in energy close to each other. (Such strong mixing between the two K π = 4 − configurations was suggested to take place in the neighboring even-Z N = 102 isotone 170 Er based on the measured gyromagnetic ratios [24].) Using the calculated energy difference (137 keV) and the mixing matrix element derived from the 170 Er case (190 keV [24]), the wave function, 67% configurations above the 4 − states in 168 Dy. The neutron 2qp configuration was assigned to the K π = 6 − isomers in the N = 102 isotones, 172 Yb [36], 170 Er [24], 166 Gd and 164 Sm [11]. Similarly to the aforementioned K = 4 − case, these two K π = 6 − components should be mixed with each other due to the proximity of their energies. It is expected that the mixed 6 − 1 state is higher in energy than the 4 − 1 state, consistent with the energy systematics shown in the upper panel of Fig. 3. The calculations also predict that the neutron 2qp excitations with K π = 6 + (ν7/2 − [514] ⊗ ν5/2 − [512]) and K π = 8 − (ν9/2 + [624] ⊗ ν7/2 − [514]) emerge at relatively low excitation energies in 170 Dy and 172 Dy, respectively. They have been identified as a long-lived isomer in each Dy isotope [13,14], while the corresponding levels were not observed in 168 Dy.
In 169 Dy, the J π = K π = (1/2 − ) state is interpreted as being predominantly of neutron 1qp nature, and its low excitation energy (166 keV) is ascribed to the rising of the ν1/2 − [521] level towards the ν5/2 − [512] one. This state becomes a long-lived isomer because, in addition to the small energy gap, the 1/2 − → 5/2 − "K -allowed" E2 transition proceeds with changes in the asymptotic quantum numbers, [ , N, n z , ] = [2, 0, 1, 1] [43]. Indeed, the corresponding E2 transition in the N = 103 isotone 173 Yb has the smallest B(E2) value observed to date in any odd-A deformed nuclei [29]. As exhibited in the lower panel of Fig. 3, however, the B(E2) values in 169 Dy and 171 Er are more than two orders of magnitude larger than that of 173 Yb. Note that the observed increase in B(E2) is in phase with a sudden fall of the γ -vibrational states in even-A Dy and Er isotopes compared to the Yb levels, indicating that γ -vibrational motion is enhanced [13]. As such, in order to explain the increase of B(E2) in 169 Dy, we consider states of mixed 1qp and γ -vibrational character. Based on the prescription described in Ref. [44], the wave functions of the 1/2 − and 5/2 − states can be described as where the |ν ⊗ 0 + gs and |ν ⊗ 2 + γ terms represent the coupling of a neutron 1qp state with the ground state and the K π = 2 + γ -vibrational excitation in the even-even core, respectively. The mixing coefficients α 1/2 , α 5/2 1 are given by the perturbation theory as The numerators are the mixing matrix elements between the coupled states, which, for simplicity, are assumed to be equal to each other and denoted by V in the following discussion. By omitting the cross term that contains the product of α 1/2 and α 5/2 in the E2 matrix element, and using the single-particle and collective transition strengths, the reduced transition probability can be approximated as The values of B(E2; ν1/2 − → ν5/2 − ) = 1.88 × 10 −2 e 2 fm 4 and E 1qp = 399 keV are taken from the experimental data of 173 Yb [29] since, as suggested in Ref. [45], the relative smallness of the E2 matrix element between these states points to an almost total absence of ν5/2 − [512] ⊗ 2 + γ in the 1/2 − band. As for the γ -vibrational core, the value of E 2 + γ is estimated to be 1150 keV with the assumption that the observed 1227-keV state in 168 Dy is the J π = 3 + member of the γ -vibrational band, while B(E2; 2 + γ → 0 + gs ) = 2.06 × 10 2 e 2 fm 4 is adopted from 170 Er [29] that shows a similar enhancement of γ vibration. In consequence, the parameters V = −34 keV, α 1/2 = 0.022, α 5/2 = 0.045 are obtained. It can be seen that a tiny (< 1%) admixture of γ vibration with the dominating neutron 1qp component enables the large increase in the E2 strengths for the 1/2 − → 5/2 − transition.
The weak mixing phenomenon observed in the present work is at variance with the majority of data concerning the occurrence of γ -vibrational mixing in well-deformed odd-A nuclides: the observed low-lying K = 1/2 states of mixed 1qp-plus-γ -vibrational character are in most cases found to contain appreciable admixtures of collective components [45]. For instance, approximately equal components of the admixed 1qp and ν ⊗ 2 + γ configurations were suggested for the K π = 1/2 − state at 707 keV in 171 Er and the K π = 1/2 + states at 581 keV in 159 Tb and at 879 keV in 183 Re.
Such strong γ -vibrational mixing is known to occur with V of the order of a few hundreds of keV, when the admixed 1qp excitation is connected to the "base" state, to which 2 + γ is coupled, by large (non-axial) quadrupole matrix elements, i.e., their asymptotic quantum numbers differ by = = ±2 with N and n z unchanged.
To conclude, the excited states of 168 Dy and 169 Dy, which have two-and one-neutron vacancies relative to the doubly midshell nucleus 170 Dy, have been populated following the decay from previously unreported isomeric states with half-lives of 0.57(7) μs and 1.26 (17) μs, respectively. In 168 Dy, the J π = K π = (4 − ) isomeric state at 1378 keV is expected to have a mixed proton and neutron 2qp configuration, while the HFB+QRPA calculation suggests that the J π = K π = (3 + ) level, which is located 40 keV below the iso-