Normal-deformed structures in hafnium isotopes

The structures of yrast normal-deformed (ND) bands in Hf isotopes have been investigated using the unpaired cranked Nilsson-Strutinsky (CNS), and the paired cranked Nilsson-Strutinsky-Bogoliubov (CNSB) models. New optimized Nilsson parameters have been derived and used in the models to interpret the experimental data in ${}^{168–175}\mathrm{Hf}$ isotopes. The calculated kinematic moments of inertia, as well as the excitation energies of the yrast bands, have been compared with the experimental findings, and good agreements are observed. Configuration changes along the yrast lines have been traced by the aid of the CNS calculations. The calculated single-particle excitations shed more light on the nature of the crossings and successfully explain some features of the observed rotational bands which had not been clear before.

Figure 1

Top panel: Unpaired CNS energies. Middle panel: Paired CNSB energies. Lower panel: Their differences for positive-parity yrast configurations in ${}^{168}\mathrm{Hf}$ (excitation energies are plotted relative to a rotating liquid drop reference).

Figure 2

Top panel: Single-neutron energies. Lower panel: Single-proton energies as functions of rotational frequency using the standard parameters at the deformation ${\varepsilon }_2\approx 0.23,\gamma \approx 0$, and ${\varepsilon }_4\approx 0.015$.

Figure 3

Top panel: Single-neutron energies. Lower panel: Single-proton energies as functions of rotational frequency using new parameters at the deformation ${\varepsilon }_2\approx 0.23,\gamma \approx 0$, and ${\varepsilon }_4\approx 0.015$.

Figure 4

Calculated kinematic moments of inertia, $J^{\left(1\right)}$, of some yrast bands in Hf isotopes using standard (Std.) and new parameters along with the experimental values versus frequency.

Figure 5

Top panel: Calculated paired and unpaired yrast bands. Middle panel: Experimental band in ${}^{168}\mathrm{Hf}$ (excitation energies are calculated relative to a rotating liquid drop energy reference). Lower panel: Calculated paired (CNSB) and experimental energy differences.

Figure 6

Calculated kinematic moments of inertia, $J^{\left(1\right)}$, for bands in Fig. 5. Inset compares the experimental and paired CNSB $J^{\left(1\right)}\mathrm{s}$.

Figure 7

Top panel: Calculated paired and unpaired yrast bands. Middle panel: Experimental bands A and B in ${}^{169}\mathrm{Hf}$ (excitation energies are calculated relative to a rotating liquid drop energy reference). Lower panel: Paired (CNSB) and experimental energy differences. Solid symbols correspond to signature $\alpha =1/2$ and open symbols to signature $\alpha =-1/2$.

Figure 8

Top panel: Calculated paired and unpaired yrast bands. Middle panel: Experimental band 1 in ${}^{170}\mathrm{Hf}$ (excitation energies are calculated relative to a rotating liquid drop energy reference). Lower panel: Paired (CNSB) and experimental energy differences.

Figure 9

Top panel: Calculated paired and unpaired yrast bands. Middle panel: Experimental bands A and B in ${}^{171}\mathrm{Hf}$ (excitation energies are calculated relative to a rotating liquid drop energy reference). Lower panel: Paired (CNSB) and experimental energy differences. Solid symbols correspond to signature $\alpha =1/2$, and open symbols correspond to signature $\alpha =-1/2$.

Figure 10

Top panel: Calculated kinematic moments of inertia, $J^{\left(1\right)}$, of yrast bands in ${}^{171}\mathrm{Hf}$ for positive signature. Lower panel: Those for negative signature. Insets compare paired and experimental $J^{\left(1\right)}\mathrm{s}$.

Figure 11

Top panel: Calculated paired and unpaired yrast bands. Middle panel: Experimental band 1 in ${}^{172}\mathrm{Hf}$ (excitation energies are calculated relative to a rotating liquid drop energy reference). Lower panel: Paired (CNSB) and experimental energy differences.

Figure 12

Top panel: Calculated paired and unpaired yrast bands. Middle panel: Experimental bands A and B in ${}^{173}\mathrm{Hf}$ (excitation energies are calculated relative to a rotating liquid drop energy reference). Lower panel: Paired (CNSB) and experimental energy differences. Solid symbols correspond to signature $\alpha =1/2$, and open symbols correspond to signature $\alpha =-1/2$.

Figure 13

Right panel: Aligned angular momenta (alignment), $i_x$, of bands A and B. Left panel: Those of paired and unpaired calculated bands in ${}^{173}\mathrm{Hf}$.

Figure 14

Top panel: Calculated paired and unpaired yrast bands. Middle panel: Experimental band 1 in ${}^{174}\mathrm{Hf}$ (excitation energies are calculated relative to a rotating liquid drop energy reference). Lower panel: Paired (CNSB) and experimental energy differences.

Figure 15

Top panel: Calculated paired and unpaired yrast bands. Middle panel: Experimental bands A and B in ${}^{175}\mathrm{Hf}$ (excitation energies are calculated relative to a rotating liquid drop energy reference). Lower panel: Paired (CNSB) and experimental energy differences. Solid symbols correspond to signature $\alpha =1/2$ and open symbols correspond to signature $\alpha =-1/2$.

Figure 16

Right panel: Aligned angular momenta (alignment), $i_x$, of bands A and B. Left panel: Those of paired and unpaired calculated bands in ${}^{175}\mathrm{Hf}$.