Low-energy dipole excitations in ${}^{20}\mathrm{O}$ with antisymmetrized molecular dynamics

Low-energy dipole (LED) excitations in ${}^{20}\mathrm{O}$ were investigated by variation after $K$ projection of $\mathrm{deformation}(\beta$)-constraint antisymmetrized molecular dynamics combined with the generator coordinate method. A low-energy $E1$ mode, which is caused by surface neutron oscillation along the prolate deformation was obtained as the $1_2^{-}$ state. Moreover, a toroidal dipole (TD) mode with vortical nuclear current was obtained as the $1_1^{-}$ state with one-proton excitation on the relatively weak deformation. The low-energy $E1$ mode is a LED excitation peculiar to neutron-rich systems that does not appear in stable oxygen isotopes, whereas the TD (vortical) mode is a LED excitation that was obtained also in ${}^{16}\mathrm{O}$ and ${}^{18}\mathrm{O}$. The TD and $E1$ modes separately appear as the $K^{\pi }=1^{-}$ and $K^{\pi }=0^{-}$ components of the deformed states, respectively, but couple with each other because of $K$ mixing, and shape fluctuation. As a result of the mixing, TD and $E1$ transition strengths are fragmented into the $1_1^{-}$ and $1_2^{-}$ states. The excited bands of $K^{\pi }=0^{+}, K^{\pi }=0^{-}$, and $K^{\pi }=1^{-}$ with cluster structures were also obtained in the energy region higher than the LED states.

Figure 1

$K$-projected energy curves of ${}^{20}\mathrm{O}$ obtained by K-VAP of $\beta$-constraint AMD plotted as a function of quadrupole deformation $\beta$. The energy curve for the $K{0}^{+}\left(\beta \right)$ bases is shown in panel (a) and those for the $K{0}^{-}\left(\beta \right)$ and $K{1}^{-}\left(\beta \right)$ bases are shown in panel (b).

Figure 2

The matter densities of $K{0}^{+}$ bases at (a) $\beta =0.32$, (b) 0.52, and (c) 0.84. The intrinsic densities are integrated along the $Y$ axis and plotted on the $Z\text{−}X$ plane. The unit is ${\mathrm{fm}}^{-2}$.

Figure 3

The same as Fig. 2 but for the (a) $K{1}^{-}\left(0.32\right)$, (b) $K{0}^{-}\left(0.44\right)$, (c) $K{1}^{-}\left(0.64\right)$, and (d) $K{0}^{-}\left(0.84\right)$ bases.

Figure 4

Energy spectra of positive-parity states in ${}^{20}\mathrm{O}$ obtained by GCM and those of experimental data. In the calculated spectra, the $K^{\pi }=0_1^{+}, K^{\pi }=0_{\text{cl,1}}^{+}$, and $K^{\pi }=0_{\text{cl,2}}^{+}$ bands are shown together with the $B\left(E2\right)$ values of in-band transitions. The experimental $B\left(E2\right)$ values are taken from Refs. [10, 11]. The unit of $B\left(E2\right)$ is $e^2{\mathrm{fm}}^4$.

Figure 5

Energy spectra of negative-parity states in ${}^{20}\mathrm{O}$ obtained by GCM and experimental negative-parity spectra. In the calculated result, spectra of the $K^{\pi }=1_1^{-}, K^{\pi }=1_{\text{cl}}^{+}$, and $K^{\pi }=0_{\text{cl}}^{+}$ bands and that of the $1_2^{-}$ state are shown together with the $B\left(E2\right)$ ($e^2{\mathrm{fm}}^4$) values of in-band transitions. For the $K^{\pi }=1_{\text{cl}}^{+}$ band, spectra on a large scale are inserted in the figure.

Figure 6

GCM amplitudes of the positive-parity states. The amplitudes calculated by squared overlap with the $K{0}^{+}\left(\beta \right)$ bases are plotted as a function of $\beta$. The results for the band-head $0_1^{+}, 0_2^{+}$, and $0_4^{+}$ states of the $K^{\pi }=0_1^{+}, K^{\pi }=0_{\text{cl,1}}^{+}, K^{\pi }=0_{\text{cl,2}}^{+}$ bands are shown in panels (a), (b), and (c), respectively.

Figure 7

GCM amplitudes of the negative-parity states. The amplitudes calculated by squared overlap with the $K{0}^{-}\left(\beta \right)$ and $K{1}^{-}\left(\beta \right)$ bases are plotted by squares and circles, respectively. Panels (a), (c), and (d) show the results for the band-head $1_1^{-}, 1_6^{-}$, and $1_9^{-}$ states of the $K^{\pi }=1_1^{-}, K^{\pi }=1_{\text{cl}}^{-}$, and $K^{\pi }=0_{\text{cl}}^{-}$ bands, respectively, and panel (b) shows the result for the $1_2^{-}$ state.

Figure 8

Density distribution of protons and those of single-particle orbits in the (a) $K{0}^{+}\left(0.32\right)$, (b) $K{0}^{+}\left(0.52\right)$, and (c) $K{0}^{+}\left(0.84\right)$ bases, which correspond to the $K^{\pi }=0_1^{+}, K^{\pi }=0_{\text{cl,1}}^{+}$, and $K^{\pi }=0_{\text{cl,2}}^{+}$ bands, respectively. The upper panels show the proton density distributions by the contour lines. In the middle and lower panels, the density of the highest neutron and proton orbits are shown with color maps, respectively, with the total proton density (contour lines). The matter densities of these bases are shown in Fig. 2.

Figure 9

The same as Fig. 8, but the results for the (a) $K{1}^{-}\left(0.32\right)$, (b) $K{0}^{-}\left(0.44\right)$, (c) $K{1}^{-}\left(0.64\right)$, and (d) $K{0}^{-}\left(0.84\right)$ bases, which correspond to the $K^{\pi }=1_1^{-}$ band, the $1_2^{-}$ state, $K^{\pi }=1_{\text{cl}}^{-}$, and $K^{\pi }=0_{\text{cl}}^{-}$ bands, respectively. The matter densities of these bases are shown in Fig. 3.

Figure 10

Dipole transition strengths for the (a) $E1$, (b) ISD, (c) TD, and (d) CD operators from the $0_1^{+}$ state. For the ISD operator, the energy-weighted strengths are plotted in ratio to the EWSR defined by Ref. [5].

Figure 11

(Upper) Transition current densities $\delta {\mathit{j}}^K\left(\mathit{r}\right)$ from the $K{0}^{+}\left(0.32\right)$ base to the $K{1}^{-}\left(0.32\right)$ corresponding to the $0_1^{+}\to 1_1^{-}$ transition and (lower) those to the $K{0}^{-}\left(0.44\right)$ base for $0_1^{+}\to 1_2^{-}$. The vector plots of the densities in the $Z\text{−}X$ plane at $Y=0$ are shown. The proton and neutron currents are shown in the left and middle, respectively, and the isovector currents are shown in the right. The vector plots are multiplied by 30 in (a)–(c), by 50 in (d) and (e), and by 100 in panel (f).

Figure 12

(Left) TD strength densities ${\mathcal{M}}_{\text{TD}}^K$ for $K{0}^{+}\left(0.32\right)\to K{1}^{-}\left(0.32\right)$ and (right) $E1$ strength densities ${\mathcal{M}}_{E1}^K$ for $K{0}^{+}\left(0.32\right)\to K{0}^{-}\left(0.44\right)$. The former and the latter correspond to the $0_1^{+}\to 1_1^{-}$ and $0_1^{+}\to 1_2^{-}$ transitions, respectively. The strength densities ${\mathcal{M}}^K(X,Y,Z)$ and $\left|X\right|$-weighted values $\left|X\right|{\mathcal{M}}^K(X,Y,Z)$ on the $Z\text{−}X$ plane at $Y=0$ are shown in upper and lower panels, respectively.

Figure 13

Energy spectra of the $0_{1,2}^{+}$ and $1_{1,2}^{-}$ states in ${}^{16}\mathrm{O}, {}^{18}\mathrm{O}$, and ${}^{20}\mathrm{O}$ calculated with K-VAP and GCM of $\beta$-constraint AMD. For excited states, intrinsic matter densities of the dominant bases are also shown with labels “TD:1p1h”, “cluster”, “cluster-doublet”, and “E1”, which indicate the TD mode with 1p-1h configuration, $K^{\pi }=0^{+}$ cluster state, its parity doublet $K^{\pi }=0^{-}$ state, and the $E1$ mode, respectively. The color map plotting of densities is the same as Fig. 2.

Figure 14

The IS, proton, and neutron components of the TD and CD strengths for the $1_1^{-}$ and $1_2^{-}$ states of ${}^{16}\mathrm{O}, {}^{18}\mathrm{O}$, and ${}^{20}\mathrm{O}$ calculated with K-VAP and GCM of $\beta$-constraint AMD. The TD strengths of (a) ${}^{16}\mathrm{O}$, (c) ${}^{18}\mathrm{O}$, and (e) ${}^{20}\mathrm{O}$ are shown in the left, and the CD strengths of (b) ${}^{16}\mathrm{O}$, (d) ${}^{18}\mathrm{O}$, and (f) ${}^{20}\mathrm{O}$ are shown in the right. Proton and neutron components are multiplied by a factor of four to compare the IS component. The results for ${}^{16}\mathrm{O}$ and ${}^{18}\mathrm{O}$ are taken from Refs. [31, 32].

Figure 15

$E1$ strengths for the $1_1^{-}$ and $1_2^{-}$ states of (a)${}^{16}\mathrm{O}$, (b)${}^{18}\mathrm{O}$, and (c)${}^{20}\mathrm{O}$ calculated with K-VAP and GCM of $\beta$-constraint AMD. The results for ${}^{18}\mathrm{O}$ are taken from Ref. [32].