Intertwined quantum phase transitions in the Zr isotopes

We explore the situation of intertwined quantum phase transitions (IQPTs), for which a QPT involving a crossing of two configurations is accompanied by a shape evolution of each configuration with its own separate QPT. We demonstrate the relevance of IQPTs to the Zr isotopes, with such coexisting Type I and Type II QPTs, and ground state shapes changing from spherical to prolate axially deformed and finally to $\gamma$-unstable. Evidence for this scenario is provided by a detailed comparison with experimental data, using a definite symmetry-based conceptual framework.

Figure 1

Comparison between (a) experimental [19, 42] and (b) calculated energy levels $0_1^{+},2_1^{+},4_1^{+},0_2^{+},2_2^{+},4_2^{+}$. Empty (filled) symbols indicate a state dominated by the normal $A$ configuration (intruder $B$ configuration), with assignments based on the decomposition of Eq. (6). Note that the calculated values start at neutron number 52, while the experimental values include the closed shell at 50.

Figure 2

Evolution of order parameters and of observables along the Zr chain. Symbols (solid lines) denote experimental data (calculated results). Relevant parameters are given in the text. (a) Order parameters (see text for details). (b) $B\left(E2\right)$ values in Weisskopf units (W.u.). Data were taken from [15, 16, 17, 18, 20, 21, 42]. Dotted lines denote calculated $E2$ transitions within a configuration. (c) Isotope shift $\mathrm{\Delta }{\langle {\widehat{r}}^2\rangle }_{0_1^{+}}$ in ${\mathrm{fm}}^2$. Data were taken from [43]. The horizontal dashed line at $0.235{\mathrm{fm}}^2$ represents the smooth behavior in $\mathrm{\Delta }{\langle {\widehat{r}}^2\rangle }_{0_1^{+}}$ due to the $A^{1/3}$ increase of the nuclear radius. (d) Two-neutron separation energies, $S_{2n}$, in MeV. Data were taken from AME2016 [44].

Figure 3

Experimental and calculated energy levels in MeV and $E2$ rates in W.u. for ${}^{100}\mathrm{Zr}$ [panels (a) and (b)] and ${}^{110}\mathrm{Zr}$ [panels (c) and (d)].

Figure 4

Contour plots in the $(\beta ,\gamma )$ plane of the lowest eigenpotential surface, $E_{-}(\beta ,\gamma )$, for the ${}^{92–110}\mathrm{Zr}$ isotopes.