Shells in a toroidal nucleus in the intermediate-mass region

Attention is fixed on shells in toroidal nuclei in the intermediate-mass region using a toroidal single-particle potential. We find toroidal shells in the intermediate-mass region with large single-particle energy gaps at various nucleon numbers. These toroidal shells are located at different toroidal deformations characterized by the aspect ratios of the toroidal major radius to the toroidal minor radius, and they provide extra stability at various toroidal deformations. Relative to a toroidal core, the Bohr–Mottelson spin-aligning particle-hole excitations may be constructed to occupy the lowest single-particle Routhian energies to lead to toroidal high-spin isomers with different spins. Furthermore, because a nucleon in a toroidal nucleus possesses a vorticity quantum number, toroidal vortex nuclei may be constructed by making particle-hole excitations in which nucleons of one type of vorticity are promoted to populate unoccupied single-particle orbitals of the opposite vorticity. Methods for producing toroidal high-spin isomers and toroidal vortex nuclei are discussed.

Figure 1

The cylindrical coordinates $(\rho ,z,\varphi )$ and toroidal coordinates $(r_{\perp },\theta ,\varphi )$ used for the description of a toroidal nucleus with a major radius $R$, a minor radius $d$, and a vorticity ${\mathrm{\Lambda }}_{\perp }$. Here, $\varphi$ is the toroidal angle, $\theta$ is the poloidal angle, and $r_{\perp }={[{(\rho -R)}^2+z^2]}^{1/2}$. Some of the poloidal vortex flow lines are schematically exhibited.

Figure 2

Single-particle energies for a toroidal nucleus as a function of the aspect ratio $R/d$. The states are labeled by quantum numbers $(n_{\rho },n_z,|{\mathrm{\Lambda }}_z|,|{\mathrm{\Omega }}_z\left|\right)$, which are displayed only for the lowest states. The locations of the toroidal shell are shown as bracketed numbers, $\left(N\right)$, at their corresponding toroidal deformations, $R/d$.

Figure 3

(a) Single-particle state energy $\epsilon$ for a toroidal nucleus as a function of the aspect ratio $R/d$. The states are labeled with quantum numbers $(n_{\rho },n_z,|{\mathrm{\Lambda }}_z|,|{\mathrm{\Omega }}_z\left|\right)$. (b) Single-particle Routhian ${\epsilon }_{\mathrm{Routhian}}$ as a function of the cranking frequency $\hslash \omega$, at the toroidal deformation $R/d=2.9$. The bracketed numbers are favorable $(N,I)$ configurations at this deformation.

Figure 4

A schematic example of how a toroidal high-spin isomer of an excited projectile may be produced by a deep-inelastic heavy-ion scattering between a projectile nucleus and a target nucleus, showing a cut in the collision plane. We display (a)–(c) in the target nucleus rest frame, and we show (d)–(g) in the excited projectile frame. The time sequence proceeds from (a) to (e). How (e) evolves to (f) or (g) depends on whether the excitation energy $E$ and spin $I=I_z$ are the same as those of the toroidal high-spin isomers.

Figure 5

A schematic example of how a toroidal vortex isomer of the excited target remnant may be produced by the punching through of a smaller projectile on a larger target nucleus in an energetic nearly-head-on collision, showing a cut in the collision plane. We display (a)–(c) in the target nucleus rest frame, while we show (d)–(e) in the excited remnant center-of-mass frame.