Pseudospin Symmetry and Microscopic Origin of Shape Coexistence in the ${}^{78}\mathrm{Ni}$ Region: A Hint from Lifetime Measurements

Lifetime measurements of excited states of the light $N=52$ isotones ${}^{88}\mathrm{Kr}$, ${}^{86}\mathrm{Se}$, and ${}^{84}\mathrm{Ge}$ have been performed, using the recoil distance Doppler shift method and VAMOS and AGATA spectrometers for particle identification and gamma spectroscopy, respectively. The reduced electric quadrupole transition probabilities $B(E2;2^{+}\to 0^{+})$ and $B(E2;4^{+}\to 2^{+})$ were obtained for the first time for the hard-to-reach ${}^{84}\mathrm{Ge}$. While the $B(E2;2^{+}\to 0^{+})$ values of ${}^{88}\mathrm{Kr}$, ${}^{86}\mathrm{Se}$ saturate the maximum quadrupole collectivity offered by the natural valence ($3s$, $2d$, $1g_{7/2}$, $1h_{11/2}$) space of an inert ${}^{78}\mathrm{Ni}$ core, the value obtained for ${}^{84}\mathrm{Ge}$ largely exceeds it, suggesting that shape coexistence phenomena, previously reported at $N\lesssim 49$, extend beyond $N=50$. The onset of collectivity at $Z=32$ is understood as due to a pseudo-SU(3) organization of the proton single-particle sequence reflecting a clear manifestation of pseudospin symmetry. It is realized that the latter provides actually reliable guidance for understanding the observed proton and neutron single particle structure in the whole medium-mass region, from Ni to Sn, pointing towards the important role of the isovector-vector $\rho$ field in shell-structure evolution.

Figure 1

Graphical determination of the lifetimes of the $2^{+}$ (a) and $4^{+}$ (b) excited states of ${}^{84}\mathrm{Ge}$ (see text). The shaded area and the two thinner curves materialize the $1\sigma$ uncertainty regions. The insets show the relevant regions of the gamma spectrum, with the shifted and unshifted photo-peak components marked with “$S$” and “$U$,” respectively.

Figure 2

$B(E2;2^{+}\to 0^{+})$ systematics of the light $N=52$ even-even isotones from $Z=38$ down to $Z=30$. “EXP. Lif.”: experimental values from lifetimes measurements (this work, except Sr [34]); “EXP. Coul.”: experimental values from Coulomb excitation measurements [31]; “DD-PC1”: beyond mean-field calculations using the relativistic functional DD-PC1 (this work); “SM,Ni78-I” shell-model calculations from [32]; “($g_{9/2}$)P-SU(3)”: pseudo-SU(3) limit [32] (or including one $N=50$ core-breaking $g_{9/2}$ pair promotion, as illustrated by the inset).

Figure 3

Relativistic-CMF proton SP energy in canonical basis for ${}^{84}\mathrm{Ge}$. Solid (blue), dashed (red), and dashed-dotted (green) curves correspond to positive parity, negative parity, and Fermi levels, respectively. Left and right panels show the evolution of the SP levels with $\beta$ at axial $\gamma =0°$ and $\gamma =60°$ limits, respectively, while the central one shows evolution with the asymmetry gamma parameter at fixed mean-field minimum ($\beta =0.2$). The pseudo-Nilsson labels $[\tilde{N}{n}_3\tilde{\mathrm{\Lambda }}]$ of the deformed states stemming from the $\tilde{s}\tilde{d}$ spherical system (dotted frame) are reported close to the relevant levels. Their evolution towards pseudospin partners degeneracy at larger deformation is shown in the inset.

Figure 4

Reduced PSO splitting ${\mathrm{\Delta }}_{\mathrm{PSO}}=({\epsilon }_{j_{<}}-{\epsilon }_{j_{>}})/[\hslash \omega (2\tilde{l}+1)]$ as a function of isospin $T$ for the $2\tilde{p}$ and $1\tilde{f}$ doublets in Sb isotopes and $N=51$ isotones and for the $1\tilde{d}$ doublet in Cu isotopes extracted from spectroscopic data [34, 41].