Shape fluctuations in the ground and excited $0^{+}$ states of ${}^{30,32,34}$Mg

Large-amplitude collective dynamics of shape phase transition in the low-lying states of ${}^{30-36}$Mg is investigated by solving the five-dimensional (5D) quadrupole collective Schrödinger equation. The collective masses and potentials of the 5D collective Hamiltonian are microscopically derived with use of the constrained Hartree-Fock-Bogoliubov plus local quasiparticle random phase approximation method. Good agreement with the recent experimental data is obtained for the excited $0^{+}$ states as well as the ground bands. For ${}^{30}$Mg, the shape coexistence picture that the deformed excited $0^{+}$ state coexists with the spherical ground state approximately holds. On the other hand, large-amplitude quadrupole-shaped fluctuations dominate in both the ground and the excited $0^{+}$ states in ${}^{32}$Mg, providing a picture that is different from the interpretation of the “coexisting spherical excited $0^{+}$ state” based on the naive inversion picture of the spherical and deformed configurations.

Figure 1

Collective potentials for ${}^{30-36}$Mg. The HFB equilibrium points are indicated by red circles.

Figure 2

Comparison of calculated excitation energies of the $2_1{}^{+}$ and $4_1{}^{+}$ states (upper panel) and $B(E2;2_1{}^{+}\to 0_1{}^{+})$ values (lower panel) in ${}^{30-36}$Mg with experimental data [1, 2, 3, 4, 5, 6].

Figure 3

Excitation energies of the excited $0_2{}^{+}$, $2_{2,3}{}^{+}$, and $4_{2,3}{}^{+}$ states (upper panel) and the ratio $B(E2;0_2{}^{+}\to 2_1{}^{+})/B(E2;0_1{}^{+}\to 2_{2,3}{}^{+})$ of the interband $E2$ transition strengths between the $K=0_2$ and $K=0_1$ bands (lower panel). Experimental data are taken from Refs. [21, 22]. See text for details.

Figure 4

Vibrational wave functions squared ${\sum }_K{\left|{\Phi }_{\alpha IK}(\beta ,\gamma )\right|}^2$ of the $0_1{}^{+},2_1{}^{+},0_2{}^{+}$, and $2_{2,3}{}^{+}$ states in ${}^{30-34}$Mg. Contour lines are drawn at every eighth part of the maximum value.

Figure 5

(a) Vibrational wave functions squared $|{\Phi }_{\alpha ,I=0,K=0}(\beta ,\gamma =0.5^{\circ }){|}^2$ of the $0_1{}^{+}$ states in ${}^{30-34}$Mg. Their values along the $\gamma =0.5^{\circ }$ line are plotted as functions of $\beta$. (b) Probability densities integrated over $\gamma$, $P\left(\beta \right)\equiv \int d\gamma |{\Phi }_{\alpha ,I=0,K=0}{(\beta ,\gamma )|}^2{\left|G(\beta ,\gamma )\right|}^{1/2}$, of the $0_1{}^{+}$ states in ${}^{30-34}$Mg, plotted as functions of $\beta$. (c) Same as (a) but for the $0_2{}^{+}$ states. (d) Same as (b) but for the $0_2{}^{+}$ states.