Shell-model study of octupole collectivity near ${}^{208}\mathrm{Pb}$

We show that the collectivity of the particle-hole wave function of low-lying octupole $3^{-}$ states in doubly magic nuclei is mainly due to the neutron-proton interaction. Both the enhanced reduced transition probability to the ground state, $B(E3;3_1^{-}\to 0_1^{+})$, and the coupling of the octupole excitation to a nucleon result from the coherent action of all the components of the collective state. The results obtained with a realistic shell-model interaction both for ${}^{208}\mathrm{Pb}$ and for ${}^{209}\mathrm{Pb}$ agree with the geometric collective model of Bohr and Mottelson, where octupole excitations are associated with phonons corresponding to collective shape oscillations of the surface of the nucleus.

Figure 1

(a) Components in the wave function of the collective $3_{\mathrm{c}}^{-}$ state calculated with a realistic shell-model Hamiltonian ($c_{k^{\prime }k}^{\rho }$; black) and compared with the analytic expression (3) (${\overline{c}}_{k^{\prime }k}^{\rho }$ with ${\alpha }_{\nu }=0.734$ and ${\alpha }_{\pi }=0.679$; blue). (b) The reduced transition matrix elements $\langle E3\rangle \equiv \langle 0_1^{+}\parallel E3\parallel j_{\rho k^{\prime }}{j}_{\rho k}^{-1};3^{-}\rangle$, in units $e{\mathrm{fm}}^3$, multiplied with the coefficients $c_{k^{\prime }k}^{\rho }$ (black) and ${\overline{c}}_{k^{\prime }k}^{\rho }$ (blue). (c) Components calculated with a realistic shell-model Hamiltonian with zero $\nu \pi$ interaction (${\tilde{c}}_{k^{\prime }k}^{\rho }$; red). On the $y$ axis are indicated all possible 1p1h configurations in a simplified notation, e.g., $f_5\to s_1$ stands for $|s_{1/2}{f}_{5/2}^{-1};3^{-}\rangle$.

Figure 2

Overlaps $|\langle J_k^{\pi }|J_{\mathrm{o}i}^{\pi }\rangle {|}^2$ (see text) for the $J^{\pi }=9/2^{+}$ and $15/2^{-}$ states in ${}^{209}\mathrm{Pb}$. The two eigenstates $|J_{\mathrm{o}i}^{\pi }\rangle , i=1,2$, are obtained in the octupole model with the single-particle orbitals $g_{9/2}$ and $j_{15/2}$, and their coupling to $3_{\mathrm{c}}^{-}$. The eigenstates $|J_k^{\pi }\rangle$ are obtained with the realistic shell-model Hamiltonian in the 1p0h$+$2p1h space. For display purposes the squared overlaps are multiplied by 1, 10, 100, or 1000, indicated in black, red, blue, or green, respectively.

Figure 3

Distribution of a neutron in the $j_{15/2}$ orbital coupled to the octupole excitation $3_{\mathrm{c}}^{-}$ of ${}^{208}\mathrm{Pb}$. The bars (in the color code of Fig. 2) indicate the overlaps $|\langle J_k^{\pi }|3_{\mathrm{c}}^{-}\times j_{15/2};J\rangle {|}^2$ for $J^{\pi }=13/2^{+},21/2^{+}$ in ${}^{209}\mathrm{Pb}$, where $|J_k^{\pi }\rangle$ are eigenstates of the shell-model Hamiltonian in the 1p0h$+$2p1h space.