General treatment of vortical, toroidal, and compression modes

Abstract

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 to the hydrodynamical vorticity is briefly discussed.

Toroidal, compressive, and E1 properties of low-energy dipole modes in Be10

“…In Ref. [21], they call the former and the latter, the toroidal and vortical modes, respectively, and described general treatment of toroidal, compressive, and vortical modes and their relation to each other. In this paper, we basically follow the descriptions of the TD, CD, and vortical dipole (VD) operators in Ref.…”

“…One of the interesting problems is whether the vorticity origin TD mode arises as low-energy resonances in nuclear systems. In works with quasiparticle phonon model (QPM) and random phase approximation (RPA) for such nuclei as 208 Pb and 132 Sn [19,[21][22][23], it has been shown that the TD mode dominates the low-energy part of the ISD strengths whereas the CD mode mainly excites the high-energy part for the ISGDR, indicating that the toroidal property is a key for low-energy dipole resonances. The TD dominant nature of the low-energy E1 resonances has been demonstrated by toroidal flow in transition current densities [1,18,19,[21][22][23].…”

“…(In some works, the word "PDR" is used to call this specific mode.) Also in IS dipole (ISD) excitations, low-energy strengths known in such nuclei as 16 O, 40 Ca, and 208 Pb [12,15,16] have been discussed in relation to an exotic dipole mode, i.e., the toroidal dipole (TD) mode [1,14,[17][18][19][20][21][22][23] The TD mode carries vorticity and its character is much different from the compressive dipole (CD) mode, which is the normal mode for the ISGDR. In this decade, IS and IV properties of low-energy dipole excitations have been intensively studied to clarify essential nature of lowenergy dipole modes [1,3,4].…”

Cluster and toroidal aspects of isoscalar dipole excitations in C12

“…One is the "toroidal mode" originally determined by the second order correction in the long-wave approximation of the transition Eλ operator in an electromagnetic field [10,32], and the other is the "vortical mode" in Ravenhall-Wambach's prescription [11]. Kvasil et al described general treatment of toroidal, compressive, and vortical modes and their relation to each other [15]. They showed that the toroidal mode is a good probe for LE-ISD excitations rather than the vortical mode, though both the TD and vortical dipole (VD) operators can measure the vorticity.…”

“…The energy of the TD mode is expected to be lower than the ISGDR energy because it conserves the nuclear density. In these years, microscopic calculations with quasiparticle phonon model (QPM) and random phase approximation (RPA) have been achieved, and toroidal natures of the LED excitations in various nuclei have been investigated [3,[12][13][14][15][16][17].…”