Lifetime measurements in ${}^{102}\mathrm{Pd}$: Searching for empirical proof of the E(5) critical-point symmetry in nuclear structure

Lifetimes of yrast and nonyrast states of ${}^{102}\mathrm{Pd}$ populated via the ${}^{92}\mathrm{Zr}$(${}^{13}\mathrm{C}$, $3n$) reaction are measured by means of the recoil distance Doppler-shift technique with a Cologne plunger coupled to a GASP spectrometer. The event-by-event data accumulated at 24 plunger distances in the range of 0.1 $\mu \mathrm{m}$ to 9 mm are analyzed using the differential decay curve method. The resulting $B\left(E2\right)$ values of the $\gamma$ transitions depopulating yrast-band members up to the $I^{\pi }=8^{+}$ state are found to deviate significantly from the corresponding predictions of the E(5) critical-point symmetry.

Figure 1

Experimental energy ratios $R=E\left(J\right)/E\left(2\right)$ (filled circles) of the ground band members of ${}^{102}\mathrm{Pd}$ and corresponding predictions (solid lines) of the E(5) symmetry and the O(6) and U(5) limits of the IBA for an infinite number of bosons.

Figure 2

Part of the projection of ring 6 ($145.4^{\circ }$) generated by gating on all $\gamma$ rays detected by ring 0 ($34.5^{\circ }$) at a distance of 29 $\mu \mathrm{m}$. Doppler-unshifted components (“stop” peaks) of the strongest $\gamma$ transitions are labeled “u”, whereas Doppler-shifted ones (“flight” peaks) are labeled “s”.

Figure 3

Five gates with different widths set on the Doppler-shifted component of the $4_1^{+}\to 2_1^{+}\gamma$ transition to check for possible gate-width dependence of the mean lifetime of the $2_1^{+}$ state. Similarly, gates of different widths were set in all flight peaks used in the present work to apply the DDCM.

Figure 4

Doppler-shifted (s) and unshifted (u) components of the $2_1^{+}\to 0_1^{+}$ transition ($E_{\gamma }=556\mathrm{keV}$) as projected in ring 6 when the gate is placed on the flight peak of the $4_1^{+}\to 2_1^{+}$ transition in ring 0. Target-to-stopper distances of (a) 2 $\mu \mathrm{m}$, (b) 12 $\mu \mathrm{m}$, (c) 17 $\mu \mathrm{m}$, (d) 36 $\mu \mathrm{m}$, (e) 54 $\mu \mathrm{m}$, (f) 68 $\mu \mathrm{m}$, (g) 89 $\mu \mathrm{m}$, and (h) 500 $\mu \mathrm{m}$ are displayed, in which the evolution from a completely stopped to a fully shifted peak is illustrated. Dashed curves indicate the fitting of the peaks from which the intensities $I$ entering Eq. [3] were derived.

Figure 5

Typical coincidence decay curves and $\tau$ plots for the $2_1^{+}$ to $8_1^{+}$ yrast band members. (a, c, e, g) The normalized intensities of the shifted (open circles) and unshifted (filled circles) components of the depopulating transition are shown with the corresponding fits from napatau (solid lines). (b, d, f, h) The associated $\tau$ plots. Statistical errors are shown by dashed lines.

Figure 6

Compilation of the weighted mean lifetimes of the yrast members (a) $2_1^{+}$, (b) $4_1^{+}$, (c) $6_1^{+}$, and (d) $8_1^{+}$ deduced in this work and their associated weighted errors. Filled symbols correspond to cases where direct gating was used, whereas open symbols correspond to indirect gating. Squares denote that the gated projection of ring 6 was used in that particular analysis. Circles denote that the gated projection of ring 0 was used instead. Each individual lifetime presented contains the statistical error from napatau. To account for any possible systematic errors, the weighted average variance of the determined lifetimes was taken as the total uncertainty (see text). The weighted mean values (solid lines) as well as the errors in the weighted mean values (dashed lines) correspond to the values listed in Table 1. Values shown for the $6_1^{+}$ and $8_1^{+}$ levels have the Doppler-shift attenuation correction incorporated.

Figure 7

Typical coincidence decay curves and $\tau$ plots for the $8_1^{-}$, $9_1^{-}$, and ${10}_1^{-}$ negative-parity sideband members are presented in (a) and (b), (c) and (d), and (e) and (f), respectively. Symbols are identical to those used in Fig. 5.

Figure 8

Compilation of the weighted mean lifetimes of the negative-parity sideband levels (a) $8_1^{-}$, (b) $9_1^{-}$, and (c) ${10}_1^{-}$ deduced in this work and their associated errors. In all cases direct gating was used. Circles, upward triangles, downward triangles, and squares correspond to projection rings 0, 1, 5, and 6, respectively. As in the case of yrast lifetimes, to determine the total uncertainty of those weighted mean lifetimes the average weighted variance was taken. Solid and dashed lines, which represent the weighted mean values and their associated errors, are listed in Table 1.

Figure 9

Complete up-to-date level scheme of ${}^{102}\mathrm{Pd}$ is shown in (b) next to the E(5) critical-point symmetry level scheme (a) and an IBA-1 fit (c). Transition strengths are normalized on the $B(E2;2_1^{+}\to 0_1^{+})$ value. In (b), the strengths of the yrast transitions (bold black arrows) are derived in this work, whereas the rest are taken from the most recent compilation of evaluated data for $A=102$ [29]. E(5) excitation energies and transition strengths are taken from [27]. The comparison is clearly does not favor the characterization of ${}^{102}\mathrm{Pd}$ as an E(5) nucleus. In addition, a general IBA-1 Hamiltonian was necessary in order to describe the yrast transition strengths well, which also contradicts the E(5) hypothesis.

Figure 10

Contours of the energy functional for the case of ${}^{102}\mathrm{Pd}$ in the $\beta$, $\gamma$ plane as obtained in our IBM-1 fit. The fact that the potential surfaces are not concentric shows that the nucleus is not spherical but, rather, characterized by some degree of triaxiality, favoring lower deformations $\beta$ on the prolate side ($\gamma =0^{\circ }$) and larger ones on the oblate side ($\gamma ={60}^{\circ }$).