Quantum Phase Transition in the Shape of Zr isotopes

The rapid shape change in Zr isotopes near neutron number $N=60$ is identified to be caused by type II shell evolution associated with massive proton excitations to its $0g_{9/2}$ orbit, and is shown to be a quantum phase transition. Monte Carlo shell-model calculations are carried out for Zr isotopes of $N=50–70$ with many configurations spanned by eight proton orbits and eight neutron orbits. Energy levels and $B(E2)$ values are obtained within a single framework in good agreement with experiment, depicting various shapes in going from $N=50$ to 70. The novel coexistence of prolate and triaxial shapes is suggested.

Figure 1

(a) $2_{1,2}^{+}$ levels, (b) $0^{+}$ levels of Zr isotopes as a function of $N$. Symbols are present theoretical results with the shape classification as shown in the legends (see the text for details). Solid lines denote experimental data [6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16]. Dashed lines connect relevant results to guide the eye. The ratio between the $4_1^{+}$ and $2_1^{+}$ levels is shown in the inset of (a) in comparison to experiment. The lowest four $0^{+}$ levels are shown for ${}^{100}\mathrm{Zr}$. (c) $B(E2;2^{+}\to 0^{+})$ values as a function of $N$. Experimental data are from [13, 20, 21, 22, 23, 24, 25]. (d) Deformation parameter ${\beta }_2$. The values by other methods are shown, too.

Figure 2

T-plots for $0_{1,2}^{+}$ states of ${}^{98,100,110}\mathrm{Zr}$ isotopes.

Figure 3

(a) Occupation numbers of protons and (b) effective single-particle energies of neutrons for selected Zr isotopes. Neutron $0g_{9/2}$ is around $-12\mathrm{MeV}$, and is not shown.

Figure 4

Levels of (a) ${}^{100}\mathrm{Zr}$ and (b) ${}^{110}\mathrm{Zr}$. Prolate, oblate, and triaxial bands are shown in red, blue, and purple, respectively. The $0_4^{+}$ state in light blue in (a) is spherical. Some large $B(E2)$ values are shown in Weisskopf units (W.u.), with rigid-triaxial-rotor values in parentheses.