Nuclear shell structure and rotational bands in ${}^{86}\mathrm{Nb}$

The high spin structures in ${}^{86}\mathrm{Nb}$ are analyzed using the cranked Nilsson-Strutinsky model. Energy versus spin curves as well as deformations and electric transition quadrupole moments are considered. It is concluded that different single-particle parameters compared with the standard values lead to better agreement with recent experimental results.

Figure 1

Mass-dependence of the $N=3$ and $g_{9∕2}$ single-neutron energies. The upper panel shows a spherical Hartree-Fock (HF) calculation using SkyrmeIII parameters and the lower panel a spherical relativistic mean field calculation using the NL3 force. The energies are plotted as a function of neutron number in the range $N=20-50$ along the $\beta$-stability line. The different energies are placed so that the mean values of the $N=3$ orbitals is constant, i.e., the absolute values on the $y$ axis have no significance.

Figure 2

Spherical single-proton (upper panel) and single-neutron energies (lower panel) calculated with a modified oscillator potential for different choices of parameters. Sets A and B were found by fitting to experimental excitation energies in ${}^{\mathrm{86,87}}{}_{\mathrm{}}\mathrm{Nb}$ and ${}^{86}\mathrm{Zr}$, respectively. Both sets A and B produce the same ordering between the $f_{5∕2}$ and $p_{3∕2}$ subshells as predicted by the HF and RMF calculations.

Figure 3

Single-proton (upper panel) and single-neutron (lower panel) energies at a typical deformation of ${}^{86}\mathrm{Nb}$ shown as a function of cranking frequency $\omega$. The energies are calculated using parameter set A. At this deformation, the relevant proton configurations are built by creating two holes below the $Z=43$ gap, indicated by open circles, and the important $N=45$ configurations are formed by placing two neutrons, indicated by closed circles above the $N=43$ gap. The configurations are labeled by the number of $N=3$ holes and $g_{9∕2}$ particles as explained in the text.

Figure 4

Combination of the more interesting $Z=41$ proton and $N=45$ neutron configurations to total configurations for ${}^{86}\mathrm{Nb}$. The proton and neutron configurations are labeled by the number of $N=3$ holes and $g_{9∕2}$ particles and by the signature for an odd number of particles as explained in the text. The maximum spin values $I_{\mathit{max}}$, calculated under the assumption that a distinction can be made between orbitals of $f_{7∕2}$ character and the other $N=3$ orbitals, are also given. Parity and signature $(\pi ,\alpha )$ of the configuration is indicated along with the spin $I_{\mathit{max}}$ of the total configuration. Positive parity configurations are shown on a shaded background. The favored configurations in each group containing the same number of $g_{9∕2}$ particles are underlined. Configuration $[1_{+}2,{05}_{+}]$ is not included since it has a different deformation as seen in Fig. 12 below.

Figure 5

Calculated total energies of ${}^{86}\mathrm{Nb}$ relative to a rigid rotor reference. Parameter set A, which has been fitted to optimize the agreement between experiment and theory, is used in the CNS calculation. Terminating states are indicated by large open circles. Dashed lines are used for configurations built from particle-hole excitations across the 50 gap. These core-excited configurations are considerably higher in energy than the valence space configurations and thus less likely to be associated with experiment.

Figure 6

Comparison between experimental (upper panel) and calculated (lower panel) energies in ${}^{86}\mathrm{Nb}$ relative to a rigid rotor reference. Only configurations with positive parity likely to be identified with experiment are displayed. Also shown are the ${16}^{+}$ and ${17}^{+}$ states which are naturally interpreted as the closed core [01,05] configurations. Closed symbols are used for signature $\alpha =0$ and open symbols for $\alpha =1$. Dotted lines indicate some of the transitions between the experimental bands. Identical symbols are used in the two panels to indicate our preferred interpretation. Dashed lines in the lower panel indicate those parts of the calculated bands not seen in experiments, according to present interpretation.

Figure 7

Potential energy surface for the positive-parity configuration $[{23}_{+},{05}_{+}]$ at $I=23\hslash$. Each lattice point has been minimized with respect to ${\epsilon }_4$-deformation. The contour line spacing is $0.1\mathrm{MeV}$.

Figure 8

The electric transition quadrupole moments $Q_t$ vs initial spin $I_i$ for the positive parity bands. Experimental values for bands 2, 3, and 4 are compared with the calculated values for some configurations.

Figure 9

Calculated deformation trajectories for the configurations considered in Fig. 6. For the trajectories drawn, the deformation decreases with increasing spin. The trajectories are drawn in steps of $\Delta I=2$ up to $I=I_{\mathit{max}}$.

Figure 10

Comparison between the experimental (upper panel) and calculated (lower panel) energies of negative parity in ${}^{86}\mathrm{Nb}$ relative to a rigid rotor reference. The different rotational bands are drawn in a similar way as the positive parity bands in Fig. 6.

Figure 11

The electric transition quadrupole moments $Q_t$ vs initial spin $I_i$ for the negative parity bands in ${}^{86}\mathrm{Nb}$. Experimental values for bands 5, 6 and 9 are compared with the calculated values for some configurations.

Figure 12

Same as Fig. 9 but for negative parity configurations of ${}^{86}\mathrm{Nb}$.

Figure 13

Comparison between the experimental (upper panel) and calculated (lower panels) energies in ${}^{86}\mathrm{Zr}$ relative to a rigid rotor reference. Only bands with positive parity are drawn. Case (A) is calculated with the same parameters as for ${}^{86}\mathrm{Nb}$ and case (B) is calculated with parameters more adapted to ${}^{86}\mathrm{Zr}$.

Figure 14

Same as in Fig. 13 but for bands with negative parity.