Lifetime measurement in neutron-rich A~100 nuclei

Lifetimes of excited states of the 98,100,102Zr nuclei were measured by using the Generalized Centroid Difference Method. The nuclei of interest were populated via neutron-induced fission of 241Pu and 235U during the EXILL-FATIMA campaign. The obtained lifetimes were used to calculate the B(E2) transition strengths and β deformation Present address: saba.ansari@cea.fr Deceased


Introduction
Nuclei in the phase transition region at A∼100 have been a topic of research in nuclear structure physics for many years. Neutron rich nuclei in this mass region show a large, stable deformation and exhibit many interesting structural phenomena. The island of quadrupole deformation appearing beyond N=60 in the A∼100 mass region has first been observed in the 1960s by S.A.E. Johansson [1] in a study of γ rays emitted by fission fragments. Shortly after, Cheifetz et al [2] identified excited states and measured lifetimes in A∼100 nuclei using spontaneous fission of 252 Cf, reporting a rotational-like behavior of neutron-rich even-even Zr, Mo, Ru and Pd isotopes, consistent with theoretical predictions of Refs [3,4].
In Zr isotopes the energy of the 2 + 1 state decreases dramatically at the transition point N=60. Experimental studies also show that for N ≥ 60, the E(4 + )/E(2 + ) ratio is larger than 3, which is characteristic of a well deformed rotor [5][6][7][8]. These studies give a direct indication towards the much speculated sudden increase in absolute transition strength from 98 Zr to 100 Zr, but due to the lack of information on the low-lying states of 98 Zr, this has not yet been fully established.
The shape change phenomenon may be explained by the strong p-n interaction between proton π1g 9/2 and neutron ν1g 7/2 subshells. The protons are excited from the predominantly filled πp 1/2 shell to the predominantly empty πg 9/2 shell [9] which leads to the decrease in the spin-orbit coupling in the neutron sector and reduces the shell gap between νg 7/2 and νd 5/2 . As the occupation of the νg 7/2 neutron orbital increases, the spin-orbit splitting increases in the proton sector, reducing the energy gap between πp 1/2 and πg 9/2 . This self-stabilizing process is responsible for the appearance of the deformation in Zr isotopes.
The monopole part of the p-n interaction causes the dramatic lowering of the 0 + 2 state (from 1.58 MeV to 0.85 MeV) as soon as two neutrons are added to the ν2d 5/2 orbital i.e., as we go from 96 Zr (empty ν2d 5/2 ) to 98 Zr [10]. The lowering of this configuration continues in 100 Zr where it becomes the 0 + 1 state of 100 Zr, while the spherical ground state of 98 Zr becomes the non-yrast 0 + 2 state (0.331 MeV) lying right above the 2 + 1 state of 100 Zr (0.212 MeV). This makes 100 Zr a perfect shape transitional point as beyond N≥60, the energy of the 0 + 2 state increases significantly and only one regular rotational band is observed for 102 Zr at low excitation energy.
A similar behavior is observed in the Sr isotopic chain where the reduced transition strengths measured in both Coulomb excitation [11] and lifetime studies [13] suggest a quantum phase transition at N=58 and shape coexistence between highly-deformed prolate and spherical structures in 98 Sr. Additionally, a low mixing between the coexisting structures in 98 Sr was determined [11,12]. In order to understand how the shape transition proceeds in the Zr isotopes, we have performed a new lifetime measurement. Such measurements are crucial to determine transition strengths which gives the systematic information on nuclear deformation and collectivity.

Lifetime measurement
The experimental set up at the Institut Laue-Langevin (ILL) Grenoble, France consisted of 8 EX-OGAM Clovers and 16 cerium doped LaBr 3 detectors arranged in a compact configuration around the 241 Pu and 235 U targets to perform γ-ray spectroscopy following neutron-induced fission. Lifetimes of the low-lying excited states of 98,100,102 Zr were obtained using the Generalized Centroid Difference 1 lifetime in 100 Zr extracted from data obtained with the 241 Pu target, as an example.
The lifetime of the 2 + 1 state of 100 Zr was measured using the GCDM which is given as [13]: and where C FEP is the centroid difference between the delayed and anti-delayed time distributions related only to the Full Energy Peaks events, C exp is the measured centroid difference, which also includes the contribution from the background events (as shown in Fig. 1), C BG is the centroid difference related only to the background, p/b is the peak-to-background ratio and PRD is the Prompt Response Difference, which describes the complete time-walk of the set-up. The first (second) background correction term of Eq. 1 is related to the feeding (decay) transition in a spectrum gated on the decay (feeding) transition, and hence at the reference energy (E re f ). Determination of the PRD curve is necessary to correct the obtained lifetime from the time-walk effect as well as to minimize the systematic errors. The complete description of the procedure to obtain the PRD curve can be found in Ref. [15].  The lifetime analysis for the 2 + 1 level of 100 Zr is described in Figs. 2 and 3. Fig. 2a shows that in a fission experiment the γ-ray spectrum includes numerous γ-ray transitions from various fission fragments as well as Compton background. Having the FATIMA array combined with EXOGAM Ge detector array provides the additional advantage of using a Ge gate to select the nucleus of interest. This further improves the p/b ratio, especially in case of a complex γ-ray spectrum where the LaBr 3 energy resolution is not sufficient to separate the transitions. The decay energy (497 keV) of the 6 + 1 →4 + 1 transition of 100 Zr was used as a Ge gate to select the nucleus of interest and in accordance with Eq. 1, the reference energy gate was applied separately to the energies of the decay (2 + →0 + , 212 keV) and feeding (4 + →2 + , 352keV) transitions. The double-gated Ge and LaBr 3 spectra can be seen in Fig. 2a for the reference energy at 212 keV (2 + 1 →0 + 1 ) and in Fig. 3a for the reference energy at 352 keV (4 + 1 →2 + 1 ). The reliability of GCDM procedure is related to its mirror symmetric character and the fact that both the feeding and decay transitions are used to evaluate the lifetime. Since the background makes a non-negligible contribution to the measured centroid centroid (Fig. 1), it is necessary to do background correction. Figs. 2b and 3b shows the background correction procedure. This is performed by: 1. calculating the centroid difference between the delayed and the anti-delayed time distribution ( C BG ) at few background positions (shown by dashed lines in Fig 2b) in the vicinity of the FEP, 2. fitting these C BG points using a polynomial function (green curve in Fig. 2b) and reading the background correction value ( C BG ) at the FEP position to correct for the lifetime as per Eq. 1.

determining the p/b ratio from the doubly-gated LaBr 3 spectrum (Figs. 2a and 3a).
The same procedure is applied when the reference energy gate is at the feeding transition and the decay transition is the FEP as shown in Fig. 3. Since the reference energy is then at the feeder of the state, the background region is different and so are the background gates. It must be noted that as the reference energy is flipped from decay to feeder, both the background curve and the PRD curve are inverted in Fig. 2b as compared to Fig. 3b. The time-walk correction is directly read from the PRD curve where, PRD E decay (E f eeder ) is same as -PRD E f eeder (E decay ). The Eqs. 1 and 2 were applied to the values listed in Fig. 2b and 3b, yielding a lifetime of 830(30) ps. The same procedure was applied for 235 U target yielding a lifetime of 850 (20) ps. Finally, an average of the lifetimes obtained from both data sets has been adopted for the 2 + 1 level of 100 Zr and is given as 840 (18) ps. The lifetime measurement of the short-lived 2 + 1 state of 98 Zr was also attempted using GCDM, however due to uncertainties in the PRD and Compton-background correction only an upper limit of 6 ps was obtained. The long lifetime of the 2 + 1 state of 102 Zr was measured using the slope method and a result of 2.91(8) ns was obtained.  Figure 2: Lifetime measurement using the GCDM for the 2 + 1 state of 100 Zr, E re f set at decay transition (2 + 1 →0 + 1 , 212 keV). The centroid difference between the delayed and anti-delayed is measured at the background points (dashed lines in (a)) which is then plotted (red points) in (b). The PRD curve (solid black curve in (b)) is shifted vertically such that the PRD at E re f is 0 ps.  Table. 1) of the 2 + 1 level in the Zr isotopic chain. The present experimental results (shown in green) are compared with the literature values [5,6,[17][18][19][20][21][22][23][24] as well as the theoretical MCSM calculations [16].

Results and Discussion
The lifetimes can be used to evaluate the B(E2, ↑) transition strengths and the β deformation parameters using the following equations [17]: and where α is the total internal coefficient, τ is the lifetime in ps and E is the transition energy in keV.
The results are presented in Table 1. The comparison of thus determined experimental deformation parameter (β) with those resulting from state-of-the art Monte Carlo Shell Model calculations [16] is shown in Fig. 4. The MCSM foresees a dramatic change in the deformation parameter at 98 Zr with a large deformation beyond N = 60. Also, the sudden decrease in the energy of the 2 + 1 level and the transition strength from 98 Zr to 100 Zr is well reproduced by the MCSM calculations. The lower limit on the deformation parameter at 98 Zr does not allow a meaningful comparison between the experimental results and the theory. Therefore further study of the lifetimes in 98 Zr is essential to establish a possible phase shape transition at N=60 in the Zr chain. Table 1: Transition strength and deformation parameters for the 2 + 1 levels in the Zr chain. "Exp" denotes to the values obtained using the lifetime measured in the present work, while "Lit" refers to earlier measurements [5,6,[17][18][19][20][21][22][23][24] (14)