General treatment of vortical, toroidal, and compression modes

The multipole vortical, toroidal, and compression modes are analyzed. Following the vorticity concept of Ravenhall and Wambach, the vortical operator is derived and related in a simple way to the toroidal and compression operators. The strength functions and velocity fields of the modes are analyzed in ${}^{208}$Pb within the random-phase approximation using the Skyrme force SLy6. Both convection and magnetization nuclear currents are taken into account. It is shown that the isoscalar (isovector) vortical and toroidal modes are dominated by the convection (magnetization) nuclear current while the compression mode is fully convective. The relation between the above concept of the vorticity and the hydrodynamical vorticity is briefly discussed.

Figure 1

Isoscalar ($T=0$) and isovector ($T=1$) vortical, toroidal, and compression dipole modes in ${}^{208}$Pb, calculated with the SLy6 parameterization. The total nuclear current $j_{\text{nuc}}$ is used. The CM is computed with the operator ${\widehat{M}}_{\text{com}}$ from Eq. (38). The lines with the arrows indicate widths and energy centroids of the low-energy and high-energy branches of isoscalar $E1$ excitations observed in the $(\alpha ,{\alpha }^{\prime })$ reaction [24, 25].

Figure 2

Calculated isoscalar ($T=0$) vortical, toroidal, and compression dipole modes in ${}^{208}$Pb. For the VM and TM, the strengths with the total $j_{\text{nuc}}$, convection $j_c$, and magnetization $j_m$ current contributions to the transition operators are shown.

Figure 3

The same as in Fig. 2 but for the $T=1$ modes.

Figure 4

Comparison of SRPA and $1ph$ strengths for the $T=0$ modes. For VM and TM, the strengths are computed with the total nuclear current $j_{\text{nuc}}$.

Figure 5

The same as in Fig. 4 but for the $T=1$ modes.

Figure 6

SRPA and $1ph$ strengths for the $E1(T=1)$ GDR. Like for other modes, the strength functions are plotted without the energy weight. The experimental width and energy [54] are shown by the horizontal line and arrow, respectively.

Figure 7

(a–b) Neutron and proton vortical velocity fields ${\vec{v}}_{\nu }^q\left(z,\rho \right)$ for the state ${\omega }_{\nu }$ = 8.3 MeV with a maximal $T=0$ vortical response. (c–d) The same for the state ${\omega }_{\nu }$ = 9.1 MeV with a maximal $T=1$ vortical response. For a better view, the velocities are amplified by the factors 50 (a–b) and 20 (c–d).

Figure 8

The same as in Fig. 7 but for the states ${\omega }_{\nu }$ = 8.7 MeV ($T=0$) and ${\omega }_{\nu }$ = 9.8 MeV ($T=1$) with a maximal toroidal strength.

Figure 9

The same as in Fig. 7 but for the states ${\omega }_{\nu }$ = 7.1 MeV ($T=0$) and ${\omega }_{\nu }$ = 8.8 MeV ($T=1$) with a maximal compression strength in the low-energy branch.