Lifetime measurements of the first 2+ states in 104,106Zr: Evolution of ground-state deformations

The ﬁrst fast-timing measurements from nuclides produced via the in-ﬂight ﬁssion mechanism are reported. The lifetimes of the ﬁrst 2 + states in 104,106 Zr nuclei have been measured via β -delayed γ -ray timing of stopped radioactive isotope beams. An improved precision for the lifetime of the 2 + 1 state in 104 Zr was obtained, τ ( 2 + 1 ) = 2 . 90 + 25 − transition probabilities of the calculations indicate a persistence of prolate deformation. 104 Zr is the most deformed neutron-rich

The first fast-timing measurements from nuclides produced via the in-flight fission mechanism are reported. The lifetimes of the first 2 + states in 104, 106 Zr nuclei have been measured via β-delayed γ -ray timing of stopped radioactive isotope beams. An improved precision for the lifetime of the 2 + 1 state in 104 Zr was obtained, τ (2 + 1 ) = 2.90 +25 −20 ns, as well as a first measurement of the 2 + 1 state in 106 Zr, τ (2 + 1 ) = 2.60 +20 −15 ns, with corresponding reduced transition probabilities of B(E2; 2 + The shape of the atomic nucleus is one of its most fundamental properties. When the nuclear shells are filled to the "magic numbers" [1,2], the nucleonic distribution is spherical. In-between these major shell closures, the nuclear shape can stabilise to a non-spherical deformed configuration. The simplest of these is the quadrupole deformation, of which the two varieties are labelled as prolate (rugby ball) and oblate (discus) shapes. The quantification of these shapes in regions of the nuclear chart which display rapid shape changes as a function of nucleon number provides stringent tests to a variety of nuclear models. They also provide essential input for developing contemporary theoretical frameworks [3].
The most abundant isotope of zirconium (N = 50) exhibits doubly magic behaviour due to the reinforcement of the pf proton shell closure at Z = 40 by the major N = 50 neutron shell closure [4,5]. A region of weak proton-neutron coupling follows [6,7] up to the nearly doubly-magic 96 Zr (N = 56) [8]. A rapid shape-phase transition (or shape coexistence) occurs across N = 60 [9][10][11][12], and from thereon a large degree of collectivity and static ground-state quadrupole deformation is manifest towards the middle of the N = 50-82 shell [12,13]. The onset of deformation has been ascribed to the proton-neutron interactions of the spin-orbit partner orbitals, π g 9/2 and ν g 7/2 [14,15], and is reinforced in N ≥ 60 nuclei by the increased occupancy of the high-j low-νh 11/2 orbitals [16][17][18][19]. Laser spectroscopy measurements have shown this ground-state deformation to be strongly prolate for 101 Zr [20]. The axial symmetry of the N > 60 Zr isotopes is at odds with its higher-Z neighbours, which exhibit triaxiality [21][22][23][24], conversely lower-Z Sr isotopes exhibit a more severe transition to strongly deformed ground-state deformations [25]. The simplicity of their axially symmetric deformation could make the N > 60 Zr isotopes a good reference case for the global understanding of this wide midshell region. Although the number of active neutrons (or holes) is maximum at the mid-shell (N = 66), the observed increase of 2 + 1 energies suggests a decrease in deformation after N = 64 [13]. This has been discussed with respect to the possibility of an N = 70 sub-shell [26] with important implications for waiting points of the astrophysical r-process.
In this Letter, the first application of β-delayed γ -ray (β-γ ) fast-timing measurements of radioactive isotope beams produced through the in-flight fission mechanism is used to measure the lifetimes of the 2 + 1 states in 104, 106 Zr. The reduced transition probabilities, B(E2; 2 + , an observable which measures the correlations between ground-and excited-states, are extracted and compared to theoretical values from projected shell model and algebraic model calculations. The quadrupole deformation of the ground-states are extracted from the B(E2; 2 + compared to a variety of nuclear mean-field calculations. The experimental investigation of 104,106 Zr was carried out at the Radioactive Isotope Beam Factory (RIBF), operated by the RIKEN Nishina Center (RNC) and the Center for Nuclear Study, University of Tokyo. A 238 U 86+ primary beam of average intensity 6.24 × 10 10 particles/s was accelerated to an energy of 345 MeV/nucleon. The in-flight abrasion fission of the beam was induced by a 555 mg/cm 2 9 Be production target situated at the entrance of the BigRIPS fragment separator [27]. Fission fragments were selected through the Bρ-E-Bρ method and identified using the TOF-Bρ-E method [28]. The data were collected using two different settings of the BigRIPS and ZeroDegree spectrometers [27], centred on the β-decay parents of 104,106 Zr. A total of 6.2 × 10 5 106 Y ions were transmitted in a wide-range and very neutron-rich setting, and 3.8 × 10 5 104 Y ions in a setting with a focus on a less exotic region [29].   Fig. 1(b) shows the 140-keV 2 + 1 → 0 + g.s. transition in 104 Zr. The delayed γ -ray energy spectrum of 106 Zr, Fig. 2(b) shows the 152 keV 2 + 1 → 0 + g.s. transition. It is noteworthy that in the prompt (red) spectrum of Fig. 2(b), the 324-keV 4 + 1 → 2 + 1 transition is clearly observed, implying strong feeding of the 2 + 1 state from the 4 + 1 state. Indeed, in both nuclei the efficiency-corrected intensities from the EURICA spectra indicate the feeding of the 2 + 1 levels is ∼50% from the 4 + 1 level and ∼20% each from two other, as yet, unassigned transitions with energies between 600-800 keV. The energy-time matrices of Figs. 1(a) and 2(a) indicate that there exist no measurable delayed structures from these transitions. Therefore, a systematic uncertainty of 50 ps has been included in the upper limit of the lifetime of the 2 + 1 state based on lifetimes of the 4 + 1 state estimated from Refs. [33][34][35]. Figs. 1(c) and 2(c) show the time-difference spectra for the 2 + 1 → 0 + g.s. transition of 104 Zr and 106 Zr, respectively. Lifetimes were extracted using two approaches. The first, fitted a single decay component to the delayed shoulder of the time-difference distribution between the limits of 2 and 10 ns, as shown by the dotted blue curve. Secondly, the solid red curve shows the result of using the convolution of the detector resolution and an exponential decay as the fit function. The resolution of the detector was assumed to be Gaussian shaped, with a width obtained from a fit to the (5 − ) → (4 − ) 159 keV transition in 102 Zr [36], which was observed to be prompt. We obtain consistent lifetimes with the two methods of τ (2 + 1 ) = 2.90 +25 −20 ns and 2.60 +20 −15 ns for 104 Zr and 106 Zr, respectively. The former is in agreement with the value in Ref. [37], 2.9(4) ns, but has a higher precision.
The B(E2; 2 + Fig. 3(a) were obtained following the prescription of Ref. [45], yielding values of 0.39(2) e 2 b 2 and 0.31(1) e 2 b 2 for 104 Zr and 106 Zr, respectively. The β 2 values shown in Fig. 3(b) were derived from the B(E2; 2 + Fig. 3(a), assuming a quadrupoloid shape and including terms to 3rd order [46]. For 104 Zr and 106 Zr, this procedure gives   The direct comparison between experimentally determined B(E2; 2 + 1 → 0 + g.s. ) values and model predictions provides significant insight to microscopic structure. In particular, the projected shell model (PSM) provides the framework in which the singleparticle shell model can be applied to deformed nuclei. The PSM calculations depicted in Fig. 3(a) [49,50,22,51] in addition to the well deformed N ≥ 60 Zr nuclei. Since the energy levels and B(E2) values of triaxial Mo and Ru isotopes are well reproduced [39], the deviation of the calculations to the observations may be attributed to the IBM-1 Hamiltonian not reflecting the axial symmetry of the zirconium nuclei for N > 60.
In summary, we have reported on lifetime measurements of 2 + 1 states in 104, 106 Zr, which show that 104 Zr is the most deformed of the neutron-rich Zr isotopes. In addition, comparison of the magnitude of the extracted deformation with the results of model calculations [47,48] indicate that these nuclei are prolate deformed. Moreover, we have demonstrated that the technique of lifetime measurements following the β-decay of RI beams is a feasible method of extracting spectroscopic information at the contemporary limits of experimentally accessible nuclei. Such techniques can be further exploited at the RIBF, and also at future projects such as FAIR [52,53].