Exactly separable version of the Bohr Hamiltonian with the Davidson potential

An exactly separable version of the Bohr Hamiltonian is developed using a potential of the form $u(\beta )+u(\gamma )/{\beta }^2$, with the Davidson potential $u(\beta )={\beta }^2+{\beta }_0^4/{\beta }^2$ (where ${\beta }_0$ is the position of the minimum) and a stiff harmonic oscillator for $u(\gamma )$ centered at $\gamma =0^{°}$. In the resulting solution, called the exactly separable Davidson (ES-D) solution, the ground-state, $\gamma ,$ and $0_2^{+}$ bands are all treated on an equal footing. The bandheads, energy spacings within bands, and a number of interband and intraband $B(E2)$ transition rates are well reproduced for almost all well-deformed rare-earth and actinide nuclei using two parameters (${\beta }_0,\gamma$ stiffness). Insights are also obtained regarding the recently found correlation between γ stiffness and the γ-bandhead energy, as well as the long-standing problem of producing a level scheme with interacting boson approximation SU(3) degeneracies from the Bohr Hamiltonian.

Figure 1

(a) Potentials in both the β and γ degrees of freedom for X(5) (top) and the Davidson potential with ${\beta }_0=0$ (middle) and ${\beta }_0=2$ (bottom). Potentials are shown for the approximate separation of variables (left) and the exact separation of the variables (right). (b) Davidson potential in the β degree of freedom for a few values of the parameter ${\beta }_0$.

Figure 2

(a) $R_{4/2}=E(4_1^{+})/E(2_1^{+})$ ratio as a function of the parameter $C$ for the ES-D solution (for different values of the Davidson parameter ${\beta }_0$), the ES-X(5) solution, and for the ${\mathrm{X}}_{\mathrm{ex}}(5)$ solution [20]. The ${\mathrm{X}}_{\mathrm{ex}}(5)$ parameter $a$ is connected to parameter $C$ of the present solution through the relation $C=(2/3)\sqrt{a}$. (b) Same as (a), but for $R_{0/2}=E(0_{\beta }^{+})/E(2_1^{+})$, corresponding to the normalized β bandhead energy. (c) Same as (a), but for $R_{2/2}=E(2_{\gamma }^{+})/E(2_1^{+})$, corresponding to the normalized γ bandhead energy. See Sec. 3a for further discussion.

Figure 3

Energy ratios (a) $R_{2\beta }$ [Eq. (8)], (b) $R_{4\beta }$ [Eq. (9)], (c) $R_{\gamma }$ [Eq. (10)], and (d) $R_{\beta \gamma }$ [Eq. (11)] as functions of the parameter $C$, for the same solutions shown in Fig. 2. See Sec. 3b for further discussion.

Figure 4

Experimental data for the same energy ratios shown in Fig. 3. For each ratio, all nuclei with $A>100$ and $R_{4/2}>3.0$ for which sufficient experimental data (taken from Ref. [41]) exist have been taken into account. The predictions for ES-D, ES-X(5), and ${\mathrm{X}}_{\mathrm{ex}}(5)$ are indicated in (a), (b), and (c). The ${\mathrm{X}}_{\mathrm{ex}}(5)$ predictions lie off scale to the right in (a) and (b). See Sec. 3b for further discussion.

Figure 5

Intraband $B(E2)$ ratios for the ground-state and β bands vs the parameter $C$ as predicted by the ES-D model (labeled by the value of the ${\beta }_0$ parameter), compared with U(5) [29], X(5) [3, 8], and SU(3) [29] predictions, as described in Sec. 3c.

Figure 6

$B(E2)$ ratios from the γ band vs the parameter $C$ as predicted by the ES-D model (labeled by the value of the ${\beta }_0$ parameter), compared with the X(5) [3, 8] and SU(3) [29] predictions, as described in Sec. 3c.

Figure 7

$B(E2)$ ratios from the β band vs the parameter $C$ as predicted by the ES-D model (labeled by the value of the ${\beta }_0$ parameter), compared with the X(5) [3, 8] and SU(3) [29] predictions, as described in Sec. 3c.

Figure 8

$B(E2)$ ratios between the γ band and β band vs the parameter $C$ as predicted by the ES-D model (labeled by the value of the ${\beta }_0$ parameter), compared with the X(5) [3, 8] predictions, as described in Sec. 3c.

Figure 9

Experimental [41] level schemes (left) compared to ES-D predictions (right) for ${}^{156}\mathrm{Gd}$ (top) and ${}^{232}\mathrm{Th}$ (bottom) using the parameter sets given in Table . $\Delta K=0$ transitions are normalized to $2_1^{+}\to 0_1^{+}$, while $\Delta K=2$ transitions are normalized to $2_{\gamma }^{+}\to 0_1^{+}$.

Figure 10

(a) Experimental data [41] for the normalized β bandhead energies $R_{0/2}=E(0_{\beta }^{+})/E(2_1^{+})$ vs the energy ratio $R_{4/2}=E(4_1^{+})/E(2_1^{+})$, compared with predictions of the the ES-D and ES-X(5) solutions, as well as the predictions of the confined β soft (CBS) solution [15, 16]. (b) Same as (a), but for the normalized γ bandheads $R_{2/2}=E(2_{\gamma }^{+})/E(2_1^{+})$. See Sec. 3e for further discussion.

Figure 11

(a) γ-stiffness coefficient $(3c){}^2$ is shown as a function of the normalized γ bandhead energy $R_{2/2}=E(2_{\gamma }^{+})/E(2_1^{+})$ predicted by the ES-D solution for different values of the Davidson parameter ${\beta }_0$. (b) Same as (a), but for the rare-earth and actinide nuclei appearing in Table . See Sec. 3f for further discussion.