Description of shape coexistence in ${}^{96}\mathrm{Zr}$ based on the quadrupole-collective Bohr Hamiltonian

Experimental data on ${}^{96}\mathrm{Zr}$ indicate coexisting spherical and deformed structures with small mixing amplitudes. Although a possible geometrical description of such a shape coexistence is implied in the contemporary discussion, it does not exist yet for ${}^{96}\mathrm{Zr}$. The observed properties of the low-lying collective states of ${}^{96}\mathrm{Zr}$ based on the geometrical collective model are investigated. The quadrupole-collective Bohr Hamiltonian with the potential having two minima, spherical and deformed, is applied. Good agreement with the experimental data on the excitation energies, $B\left(E2\right)$, and $B\left(M1\right)$ reduced transition probabilities is obtained. It is shown that the low-energy structure of ${}^{96}\mathrm{Zr}$ can be described in a satisfactory way within the geometrical collective model with a potential function supporting shape coexistence without other restrictions of its shape. However, the excitation energy of the $2_2^{+}$ state can be reproduced only if the rotation inertia coefficient is taken to be 5 times smaller than the vibrational one in the region of the deformed well. It is shown also that shell effects are important for the description of $B(M1;2_2^{+}\to 2_1^{+})$. An indication of the influence of the pairing vibrational mode on the $0_2^{+}\to 0_1^{+}$ transition is obtained.

Figure 1

The wave functions of the $0_1^{+}$, $0_2^{+}$, $2_1^{+}$, and $2_2^{+}$ states.

Figure 2

Experimental (a) and calculated (b) low-lying $0^{+}$ and $2^{+}$ states of ${}^{96}\mathrm{Zr}$. Excitation energies are given in keV. The values of electric transition probabilities are given in Weisskopf units and those of magnetic ones are in nuclear magnetons. Experimental data are taken from Ref. [28].

Figure 3

The potential energy $V\left(\beta \right)$ and the calculated energy levels.