Establishing the Maximum Collectivity in Highly Deformed $N=Z$ Nuclei

The lifetimes of the first excited $2^{+}$ states in the $N=Z$ nuclei ${}^{80}\mathrm{Zr}$, ${}^{78}\mathrm{Y}$, and ${}^{76}\mathrm{Sr}$ have been measured using the $\gamma$-ray line shape method following population via nucleon-knockout reactions from intermediate-energy rare-isotope beams. The extracted reduced electromagnetic transition strengths yield new information on where the collectivity is maximized and provide evidence for a significant, and as yet unexplained, odd-odd vs even-even staggering in the observed values. The experimental results are analyzed in the context of state-of-the-art nuclear density-functional model calculations.

Figure 1

Particle identification of reaction products from the (a) ${}_{40}{}^{81}\mathrm{Zr}$ and (b) ${}_{39}{}^{80}\mathrm{Y}$ secondary beams. The plot shows energy loss against time of flight measurements from the detection system of the S800 spectrograph focal plane.

Figure 2

Doppler-shift corrected $\gamma$-ray spectra for (a) ${}^{80}\mathrm{Zr}$, (b) ${}^{78}\mathrm{Y}$, (c) ${}^{78}\mathrm{Sr}$, and (d) ${}^{76}\mathrm{Sr}$ at the specified after-target velocity ($\beta$). Experimental spectra are compared to the optimum simulated spectra (red lines) with an additional exponential background contribution (dashed blue lines). Inset plots display the ${\chi }^2$ per number of degrees of freedom (NDF) distributions as a function of mean lifetime at the optimum energy determined through the 2D ${\chi }^2$-minimization procedure employed (see text), with the minima corresponding to the measured lifetime.

Figure 3

Schematics of the $B(\mathrm{E}2\downarrow )$ values for the $N=Z$ nuclei from ${}^{64}\mathrm{Ge}$ to ${}^{80}\mathrm{Zr}$. Red symbols correspond to the present work, where ${}^{76}\mathrm{Sr}$ (triangle) is a weighted average of the present work and Ref. [11]. The black data points are taken from [5, 6, 7, 8, 9, 10] with weighted averages used where possible. The blue dashed line is to guide the eye. The inset displays $\mathrm{CHFB}+5\mathrm{DCH}$ calculations from Ref. [1] (blue dotted line) and data for the even-even $N=Z$ nuclei.

Figure 4

The left-hand panel displays the calculated Nilsson model single-proton energies against total quadrupole moment (total ${\mathrm{Q}}_{20}$). The circled numbers correspond to gaps between orbitals for a given number of protons. Since ${}_{39}{}^{78}\mathrm{Y}$ has an equal number of neutrons and protons, the single-neutron energies are very similar. The Fermi level of ${}_{39}{}^{78}\mathrm{Y}$ is at the crossing point of the [422]5/2 and [301]3/2 levels. The effects of blocking each of these orbitals on the $B(\mathrm{E}2\downarrow )$ calculated through HFB calculations (see text) is displayed in the right-hand panel.