Toroidal, compressive, and $E1$ properties of low-energy dipole modes in ${}^{10}\mathrm{Be}$

We studied dipole excitations in ${}^{10}\mathrm{Be}$ based on an extended version of the antisymmetrized molecular dynamics, which can describe 1p-1h excitations and large amplitude cluster modes. Toroidal and compressive dipole operators are found to be good proves to separate the low-energy and high-energy parts of the isoscalar dipole excitations, respectively. Two low-energy $1^{-}$ states, the toroidal dominant $1_1^{-}$ state at $E\sim 8$ MeV and the $E1$ dominant $1_2^{-}$ state at $E\sim 16$ MeV, were obtained. By analysis of transition current densities, the $1_1^{-}$ state is understood as a toroidal dipole mode with exotic toroidal neutron flow caused by rotation of a deformed ${}^6\mathrm{He}$ cluster, whereas the $1_2^{-}$ state is regarded as a neutron-skin oscillation mode, which are characterized by surface neutron flow with inner isoscalar flow caused by the surface neutron oscillation against the $2\alpha$ core.

Figure 1

Energy-weighted dipole strengths $E\tilde{B}\left(\mathrm{CD}\right),E\tilde{B}\left(\mathrm{TD}\right), E\tilde{B}\left(\mathrm{VD}\right)$, and $EB\left(E1\right)$ for the CD, TD, VD, and E1 modes of ${}^{10}\mathrm{Be}$ calculated with the $\mathrm{sAMD}+\alpha \mathrm{GCM}$. The proton and neutron contributions in the CD and TD modes are shown in (b) and (c). The smearing width is $\gamma =1$ MeV.

Figure 2

Intrinsic density distributions of protons and neutrons in ${}^{10}\mathrm{Be}\left(0_1^{+}\right)$ and ${}^{10}\mathrm{Be}\left(1_1^{-}\right)$ obtained with the AMD+VAP calculation. Density integrated along the $Y$ axis is plotted on the $X-Z$ plane.

Figure 3

Squared overlap of $\mathrm{\Psi }\left(J_k^{\pi }\right)$ with the $\alpha \mathrm{GCM}$ basis wave functions $P_{M0}^{J\pi }{\mathrm{\Phi }}_{\mathrm{AMD}}\left({\mathit{Z}}_{\alpha }^0\left(\mathrm{\Delta }D\right)\right)$. The overlaps for $\mathrm{\Psi }\left(0_1^{+}\right)$ and $\mathrm{\Psi }\left(1_2^{-}\right)$ are plotted as functions of the ${}^6\mathrm{He}-\alpha$ distance ($D=D_0+\mathrm{\Delta }D$).

Figure 4

Vector plots of the transition current densities for $|0_{1,\mathrm{int}}^{+}\rangle \to |1_{1,\mathrm{int}}^{-}\rangle$. (a) IS, (b) IV, (c) proton, and (d) neutron transition current densities ($c{\mathrm{fm}}^{-3}$ unit) at $Y=0$ are plotted on the $X-Z$ plane (scaled by a factor of ${10}^3$). Red solid (magenta dashed) lines in (a) and (b) show contours for the matter density $\rho (X,0,Z)=0.08{\mathrm{fm}}^{-3}$ of $|0_{1,\mathrm{int}}^{+}\rangle$ ($|1_{1,\mathrm{int}}^{-}\rangle$), and those in (c) and (d) show contours for the proton and neutron densities ${\rho }_{p,n}(X,0,Z)=0.04{\mathrm{fm}}^{-3}$, respectively.

Figure 5

Same as Fig. 4 but for $|0_{1,\mathrm{int}}^{+}\rangle \to |1_{2,\mathrm{int}}^{-}\rangle$.

Figure 6

Schematic figures of (a) ${}^{10}\mathrm{Be}\left(0_1^{+}\right)$, (b) ${}^{10}\mathrm{Be}\left(1_1^{-}\right)$, and (c) ${}^{10}\mathrm{Be}\left(1_2^{-}\right)$, (d) transition current in $0_1^{+}\to 1_1^{-}$, and (e) that in $0_1^{+}\to 1_2^{-}$.

Figure 7

Same as Fig. 4 but calculated for $C_{\mathrm{T}}^{+}\to C_{\mathrm{R}}^{-}$ with the ${}^6\mathrm{He}+\alpha$ cluster model at $D=3$ fm. The current densities ($c{\mathrm{fm}}^{-3}$ unit) are scaled by a factor of 500.

Figure 8

Same as Fig. 4 but calculated for $C_{\mathrm{T}}^{+}\to C_{\mathrm{T}}^{-}$ with the ${}^6\mathrm{He}+\alpha$ cluster model at $D=3$ fm. The current densities ($c{\mathrm{fm}}^{-3}$ unit) are scaled by a factor of 500.

Figure 9

$\left|K\right|=1$ component of (a) TD and (b) CD transition densities $\left[(-i){\mathrm{cfm}}^{-1}\right]x$ at $Y=0$ on the $X-Z$ plane for $C_{\mathrm{T}}^{+}\to C_{\mathrm{R}}^{-}$.

Figure 10

Decomposition of the $\left|K\right|=1$ component of the transition current densities for $C_{\mathrm{T}}^{+}\to C_{\mathrm{R}}^{-}$. Non-weighted transition current densities, (a) $j_X$ and (b) $j_Z$ ($c{\mathrm{fm}}^{-3}$), and the weighted transition current densities, (c) $X^2{j}_X$, (d) $XZ{j}_Z$, and (e) $Z^2{j}_X$ ($c{\mathrm{fm}}^{-1}$), at $Y=0$ are plotted on the $X-Z$ plane.

Figure 11

$K=0$ component of (a) TD and (b) CD transition current densities $\left[(-i){\mathrm{cfm}}^{-1}\right]$ at $Y=0$ on the $X-Z$ plane for $C_{\mathrm{T}}^{+}\to C_{\mathrm{T}}^{-}$.

Figure 12

Decomposition of the $K=0$ component of the transition current densities for the transition $C_{\mathrm{T}}^{+}\to C_{\mathrm{T}}^{-}$. Nonweighted transition current densities, (a) $j_X$ and (b) $j_Z$ ($c{\mathrm{fm}}^{-3}$), and the weighted transition current densities, (c) $ZX{j}_X$, (d) $X{X}^2{j}_Z$, and (e) $ZZ{j}_Z$ ($c{\mathrm{fm}}^{-1}$), at $Y=0$ are plotted on the $X-Z$ plane.

Figure 13

Schematic figures for transition current densities and their contributions to TD, CD, and $E1$ strengths. (a),(b) Contributions of the toroidal current to $K=1$ component of the TD and CD modes in $0_1^{+}\to 1_1^{-}$. (c),(d),(e) Contributions of the translational current to $K=0$ component of the TD, CD, and $E1$ modes in $0_1^{+}\to 1_2^{-}$.