Anharmonicity of multi–octupole-phonon excitations in ${}^{208}\mathrm{Pb}$: Analysis with multireference covariant density functional theory and subbarrier fusion of ${}^{16}\mathrm{O}+{}^{208}\mathrm{Pb}$

We discuss anharmonicity of the multi–octupole-phonon states in ${}^{208}\mathrm{Pb}$ based on a covariant density functional theory by fully taking into account the interplay between the quadrupole and the octupole degrees of freedom. Our results indicate the existence of a large anharmonicity in the transition strengths, even though the excitation energies are similar to those in the harmonic limit. We also show that the quadrupole-shape fluctuation significantly enhances the fragmentation of the two–octupole-phonon states in ${}^{208}\mathrm{Pb}$. Using those transition strengths as inputs to coupled-channels calculations, we then discuss the fusion reaction of ${}^{16}\mathrm{O}+{}^{208}\mathrm{Pb}$ at energies around the Coulomb barrier. We show that the anharmonicity of the octupole vibrational excitation considerably improves previous coupled-channels calculations in the harmonic-oscillator limit, significantly reducing the height of the main peak in the fusion barrier distribution.

Figure 1

The total energy (normalized to the GCM ground state) of the ${}^{208}\mathrm{Pb}$ nucleus as a function of octupole-deformation parameter ${\beta }_3$. The quadrupole degree of freedom is frozen at ${\beta }_2=0$. The energy curve with a projection onto the particle numbers $\left(N,Z\right)$ and the spin parity of $0^{+}$ is also shown by the solid line, together with the excitation energies and the collective wave functions for the states.

Figure 2

The low-lying energy spectra of ${}^{208}\mathrm{Pb}$ obtained by mixing the octupole-quadrupole- $({\beta }_3-{\beta }_2$-) deformed configurations [panel (b)] and by mixing only the octupole- $({\beta }_3$-) deformed configurations [panel (c)] in comparison with the experimental data taken from Ref. [38] [panel (a)]. Panels (d) and (e) show the $E3$-transition strengths from the ground state to the $3^{-}$ states and from the first $3^{-}$ state to the $2^{+}$ states, respectively, as a function of the excitation energy of the final states. In panels (a)–(c), the red solid and the blue dashed lines indicate the $E3$- and the $E2$-transition strengths (in W.u.), respectively. All the calculated excitation energies are scaled to the empirical excitation energy of the lowest $3^{-}$ state by dividing them by a constant factor.

Figure 3

The distribution of the collective wave functions $g^{J\pi }$ for several selected low-lying states of ${}^{208}\mathrm{Pb}$ shown in Fig. 2.

Figure 4

The fusion cross sections (upper panel) and the fusion barrier distributions (lower panel) for the ${}^{16}\mathrm{O}+{}^{208}\mathrm{Pb}$ system obtained with the semimicroscopic coupled-channels calculation with the coupling strengths from the MR-DFT calculations (the solid line). The dashed and the dotted lines show the results of the two-phonon coupling in the harmonic-oscillator limit and of the no-coupling limit, respectively. The experimental data are taken from Ref. [22].