Gross shell structure at high spin in heavy nuclei

Experimental nuclear moments of inertia at high spins along the yrast line have been determined for a large number of nuclei. They are found to systematically differ from the rigid-body values. The difference is attributed to shell effects which are well described by a rotating Woods-Saxon potential. The data and quantal calculations are interpreted by means of the semiclassical periodic orbit theory. From this new perspective, features in the moments of inertia as a function of neutron number and spin as well as their relation to the shell energies can be understood. Gross shell effects persist up to the highest angular momenta observed.

Figure 1

Yrast lines for a rotational nucleus ${}^{160}\mathrm{Er}$ (top) and for a nonrotational nucleus ${}^{150}\mathrm{Dy}$ (bottom) together with their fit for the ten highest spins.

Figure 2

Yrast lines for a selection of nuclei relative to their fit for the ten highest spins: ${}^{160}\mathrm{Er}$ (diamonds), ${}^{168}\mathrm{Hf}$ (squares), ${}^{170}\mathrm{Hf}$ (triangles), and ${}^{214}\mathrm{Ra}$ (crosses).

Figure 3

Experimental moments of inertia as a function of neutron number for small and normal deformation. The symbols shown at right indicate the proton number and are the same for all following figures except Fig. 5.

Figure 4

Experimental deviations from the rigid-body moments of inertia as a function of neutron number for small and normal deformation.

Figure 5

(Color online) Experimental deviations from the rigid-body moments of inertia as a function of neutron number for superdeformation.

Figure 6

Experimental shell energies at spin 0 as a function of neutron number.

Figure 7

Experimental shell energies at spin 20 as a function of neutron number.

Figure 8

Maximum angular momentum observed in each nucleus as a function of neutron number.

Figure 9

Same as Fig. 4, but for the subset of nuclei with maximum spin greater than $20\hslash$.

Figure 10

Calculated deviations of the moment of inertia from the rigid-body value as a function of neutron number for different shapes (oblate—left panels and prolate—right panels) and different orientations of the rotational axis with respect to the symmetry axis (perpendicular—lower panels and parallel—upper panels).

Figure 11

Calculated deviations of the moment of inertia from the rigid-body value as a function of the neutron number $N$ for optimal orientation of the rotational axis.

Figure 12

Calculated shell energies at spin 30 as a function of the neutron number $N$.

Figure 13

Classical orbits in the equator plane (upper panel) and the meridian plane of a normally deformed spheroidal cavity.

Figure 14

Classical orbits in the equator plane (upper panel) and the meridian plane of a superdeformed spheroidal cavity.

Figure 15

Shell energy at zero rotational frequency for two Woods-Saxon potentials. Upper panel: Cavitylike with small diffuseness $a=0.05\text{fm}$, no spin-orbit potential. Lower panel: Realistic parameters from Table and $\left(N,Z\right)=\left(104,78\right)$, with spin-orbit potential. For each $N$ we use $Z=\text{int}\left[N\left(78∕104\right)\right]$. The lines of constant action for the cavity are included. The downsloping lines show the rhombi in the meridian plane. The upsloping lines starting at $\alpha =-0.2$ show the squares in the equator plane and the upsloping lines starting at $\alpha =0.2$ show the five-point star in the equator plane. The dashed lines correspond to maxima and the full lines to minima of the respective shell energies. The values for the action are chosen such that for the equator and meridian orbits the lines go through the minima or maxima of the shell energy at $\alpha =0$. In the upper panel, the action for the five-point star orbit is chosen such that $L_{\star }{k}_F+{\nu }_{\star }=\pi \left(2n+1∕2\right)$, $n$ integer, i.e., the sine function in Eq. (20) is equal to one. In the lower panel a constant is added, which is chosen such that the magic number $N=88$ for superdeformed shape falls half way between two lines at $\alpha =0.6$.

Figure 16

Modulation factors as functions of the rotational flux $\Phi$. The dashed lines show the quadratic approximation.

Figure 17

Shell energy at finite rotational frequency for a cavitylike potential rotating perpendicular to the symmetry axis. Upper panel: $\hslash \omega =0.3\text{MeV}$, lower panel: $\hslash \omega =0.6\text{MeV}$. The lines of constant action are the same as in Fig. 15, upper panel.

Figure 18

Shell energy at finite rotational frequency for a realistic Woods-Saxon potential with spin-orbit coupling. Left panels: $\hslash \omega =0.3\text{MeV}$. Right panels: $\hslash \omega =0.6\text{MeV}$. Upper panels: rotation parallel to the symmetry axis. Lower panels: rotation perpendicular to the symmetry axis. The lines of constant action are the same as in Fig. 15, lower panel.

Figure 19

POT level density (arbitrary units) generated by the family of tetragonal orbits in a rotating spheroidal cavity. Only terms of first order in $\alpha$ are taken into account in orbit length.