Spin-parity assignments and extension of the 0 + 2 band in 158

Low and medium spin collective structures in Er have been studied using the Sm(C,4nγγ) fusion-evaporation reaction at a beam energy of Elab = 65 MeV. A band built on the 0 + 2 excitation has been established and extended to J = 18 from the analysis of γ-γ coincidence relationships, intensity arguments and DCO ratios. The 0 2 band in Er presents a similar trend to the 0 2 bands in the lighter N = 90 isotones but lies about 125 keV higher. This systematic trend supports a similar configuration for the 0 2 bands in the N = 90 isotones.


Motivation
The interpretation of excited 0 + bands built well below the pairing gap remains a fundamental open question in nuclear-structure physics.The lowest-lying 0 + 2 bands occur in the N = 90 isotones and have long been associated with the text-book picture of a β vibration, i.e. a quadrupole vibration about the nuclear surface while maintaining axial symmetry [1].The nuclides 152  62 Sm 90 and 154  64 Gd 90 may be the best examples of a β vibration [2].Nevertheless, two-phonon quadrupole vibrations have not been found in 152 Sm [3], which questions the origin of the 0 + 2 bands [3,4] and any critical-point interpretation [5][6][7].Shape coexistence has been suggested in 152 Sm [8] and 154 Gd [9] as a result of proton pairs excited across the Z = 64 midshell.The resulting proton-neutron interaction may then lower the energy of the excited 0 + bands [10,11].The large E0 strengths between the two K = 0 bands in 152 Sm, 154 Gd and 156 66 Dy 90 support the strong mixing between coexisting bands [10,12,13].This interpretation is, however, not consistent with the negligible population of the 0 + 2 state in 152 Sm observed in the ( 3 He,n) reaction study of Alford and co-workers [14].
An alternative interpretation arises from the interpretation of the 0 + 2 state as a second vacuum [4]; where these 0 + 2 excitations are explained as a two-neutron twohole seniority zero state placed into the pairing gap by the configuration-dependent pairing interaction and the low a e-mail: tdinoko@tlabs.ac.za  [4,15].This is the "pairing isomer" concept of Ragnarsson and Broglia [16] and the two-neutron character of the 0 + 2 states in 152 Sm and 154 Gd is supported by the available (t, p) transfer data [17,18].Evidence for weakly-deformed pairing isomeric bands has been found in 152 Sm [8] and 154 Gd [9], but associated with the 0 + 3 excitation.Experimentally, the second-vacuum picture is based on the congruence of similar level schemes built on the 0 + 1 and 0 + 2 vacuum states [4].The lack of experimental matrix elements prevents, however, additional support.Theoretical calculations that would accomodate such extreme lowering in the energy of the 0 + excitations into the pairing gap are lacking for either the shape-coexistence or secondvacuum pictures.The 0 + 2 bands have undoubtly been found in the stable N = 90 isotones lighter than 158  68 Er 90 [8,[19][20][21] and in 160  70 Yb 90 [22].The nucleus 158 Er has been studied through electron-capture decay [23] and heavy-ion fusionevaporation reactions [24][25][26][27].The 0 + 2 band has only been proposed up to the 4 + level [28].In this work, we present the firm assignment of previously proposed members of the 0 + 2 band in 158 Er and its extension from γ − γ coincidence techniques and discuss the available energy systematics in the N = 90 isotones.

Experimental Details
The nucleus 158 Er has been studied at the iThemba Laboratory for Accelerator Based Sciences using the 150 Sm( 12 C,4nγγ) 158  delivered by the K = 200 Separated Sector Cyclotron and bombarded a 1 mg/cm 2 150 Sm target, backed on a thick 12 mg/cm 2 Au foil.The 12 C ions are the lightest beams utilised to study 158 Er in fusion-evaporation reactions.The γ decays from the reaction products have been detected using the AFRODITE γ-ray spectrometer [29] equipped with nine escape-suppressed clover detectors; five positioned at 90 • and four positioned at 135 • .A time window of 110 ns between two γ-rays being detected by two separate clovers in the AFRODITE array characterised the γ-γ coincidence events.An average beam current of 15 enA was used and a total of about 4.2×10 8 coincidence events were accumulated during approximately fifty hours of beam time.
In the offline analysis, γ-γ coincident events were unfolded from the raw data and replayed into Radwareformat [30] for subsequent analysis.The total projection of the 150 Sm( 12 C,4nγγ) 158 Er γ-γ coincidence matrix confirms that 158 Er is the main channel in the reaction.In addition, there were open reaction channels from 5n and 3n neutron evaporations, the break-up of the carbon beam and from reactions with the Au backing.

Data Analysis
Figure 1 shows a partial decay scheme built in this work from the γ-γ coincidence data.Band 1 is the ground-state band, which continues above the S band crossing near spin 12 +  1 and band 2 is the yrast (π,α)=(+,0) S band, a sequence of 8 rotational states up to the 26 + 1 level at 7278 keV.Band 3 is the band built on the 0 + 2 excitation, which was previously known up to spin 4 + 2 [23], but has been assigned and extended in this work.The confirmation of in-band transitions for the 0 + 2 band is supported by the two γ-γ coincidence spectra shown in Fig. 2, where gates set on the 430 and 469 keV γ-ray transitions allow an arrangement of the band, which has been extended to spin 18 +  3 .The ordering of the γ-rays is also supported by the agreement of the in-band and out-of-band decays.The 1066 keV γ-ray  transition was previously observed but placed in a different band.The spins and parities of the states above the 4 + 2 level in band 3 have been firmly assigned in this work (as discussed below) and the transitions linking states between 18 + 2 and 10 + 2 are new.In this work, spin and parity values were determined using the method of directional correlations from oriented states (DCO) [31,32].The R DCO values have been obtained by gating only on stretched quadrupole transitions below the transitions of interest.∆J = 2 and pure (δ = 0) ∆J = 1 transitions are expected to have R DCO values of ≈ 1 and ≈ 0.6, respectively.For all transitions in Band 1, including the triplet of 648, 651 and 652 keV γ-ray transitions, values of R DCO ≈ 1 have been determined in agreement with available experimental data.Gamma-ray spectra gated on the 430 keV 8 + 2 → 6 + 2 transition depopulating the 0 + 2 band present similar intensities at θ lab = 90 • and 135 • clover detection angles and suggest a predominant E2 character for the in-band transitions.The R DCO values for these in-band transitions are listed in Table 1.

Discussion and Conclusions
Figure 3 shows the energy systematics of available 0 + 1 and 0 + 2 bands in the N = 90 isotones.The 0 + 2 excitation in 158 Er lies at 806.4 keV, about 125 keV higher than in the lighter N = 90 isotones.As shown in Fig. 3, the 0 + 2 band in 158 Er follows a similar trend to its counterparts in the lighter N = 90 isotones.The E(J + 2)/E(J) energy ratios for the 0 + 2 bands in the N = 90 isotones are very similar, except for a slightly smaller E(4 + 2 )/E(2 + 2 ) ratio in 158 Er and 160 Yb.This systematic trend may suggest a similar structure for the 0 + 2 bands related to a two-particle (2p) two-hole (2h) neutron configuration [17,18].In 152 Sm, 154 Gd and 156 Dy, the similarity in energy spacing between the ground and 0 + 2 band, together with the large E0 strengths between the two K = 0 bands, has been associated with strong mixing of coexisting bands with different deformations [10].In contrast, as shown in Fig. 3, the similarity in energy spacing does not continue in the heavier isotones, which indicates a reduction in deformation of the ground-state bands of 158 Er and 160 Yb, which is not mirrored in the 0 + 2 bands.Summarizing, a detailed spectroscopic study at low and medium spins in 158 Er has been carried out using the 150 Sm( 12 C,4nγγ) reaction.The spin and parities of the band built on the first excited 0 + 2 state have been assigned and the band extended to 18 +  3 from the analysis of coincidence relationships, intensity arguments and the assignment of DCO ratios.A more detailed study of the positiveand negative-parity states as well as comparison with theoretical calculations using density-functional theory will be discussed in a separate paper.Beyond mean-field calculations of Bender and Heenen [33] consider multiple np-nh excitations and may be crucial to elucidate 0 + excitations, in general.Further ( 3 He, nγ) coincidence measurements as well as multi-step Coulomb-excitation studies using stable N = 90 beams at iThemba LABS will shed light into the various interpretations argued in the N = 90 isotones.

1 Figure 3 :
Figure 3: Excitation energy as a function of angular momentum for members of the 0 + 1 and 0 + 2 bands in the N = 90 isotones.