Hartree-Fock calculations (no pairing)

Before discussing the SD bands in nuclei around ⁶⁰Zn, we briefly present some generic features of the corresponding single-particle spectra. The HF neutron single-particle orbitals near the SD N=Z=30 magic gap, calculated in ⁶⁰Zn, are shown in Fig. 1. For protons the corresponding Routhian diagrams are almost identical apart from a uniform shift in energy. The single-particle spectra show large gaps at N(Z)=30 that are stable up to the highest frequencies. At the bottom of the SD magic gap there appear two strongly deformation-driving intruder orbitals [440]1/2(r=$\pmi$), that originate from the N₀=4 harmonic oscillator (HO) shell, or more specifically, from the spherical 1g_9/2 subshell, and therefore are denoted as 4¹$\equiv$[440]1/2(r=-i) and 4²$\equiv$[440]1/2(r=+i). Above the gap, one can see six low-lying orbitals, i.e., the next two intruder states 4³$\equiv$[431]3/2(r=-i) and 4⁴$\equiv$[431]3/2(r=+i), as well as four negative-parity orbitals which in the present study are denoted as $f_\pm$$\equiv$[303]7/2(r=$\pmi$) and $p_\pm$$\equiv$[310]1/2(r=$\pmi$). The $f_\pm$ orbitals are in fact the hole states originating from the 1f_7/2 spherical subshell, while the $p_\pm$ orbitals are strong mixtures of the 1f and 2pspherical subshells, i.e., symbol $p_\pm$ is assigned only to fix a convenient naming convention.

  

Figure 1: Hartree-Fock neutron single-particle Routhians in the SD doubly magic configuration 4²4² of ⁶⁰Zn, calculated for the Skyrme interactions SLy4. Lines denoting the four (parity, signature) combination are: long-dashed (+,+i), solid (+,-i), short-dashed (-,+i), and dotted (-,-i). Standard Nilsson labels are determined by finding the dominating Nilsson components of the HF wave-functions at low (left set) and high (right set) rotational frequencies.

The doubly-magic SD configuration in ⁶⁰Zn [9], denoted by 4²4², corresponds to occupying all orbitals below the N=Z=30 gaps, and leaving empty all those that are above these gaps. Similarly, following the assignments of configurations proposed for experimentally observed bands, we have calculated three other SD bands, for the 4¹4¹ (⁵⁸Cu [28]), 4²4¹ (⁵⁹Cu [11]), and 4³4² (⁶¹Zn [10]) configurations. The relative alignments (i.e., differences of angular momenta at fixed rotational frequencies) with respect to the SD band in ⁵⁸Cu are shown in Fig. 2. Since the experimental SD bands in ⁵⁹Cu, ⁶⁰Zn, and ⁶¹Zn extend to higher rotational frequencies than that in ⁵⁸Cu, we have artificially extended the latter band by adding two gamma rays at 3641 and 4128 keV. This was done for the presentation purpose only; alternatively, we could have used the ⁵⁹Cu band as the reference, however, this would have not allowed us to show the relative alignments at lower rotational frequencies. Since the exit spins of the ⁵⁸Cu and ⁶¹Zn bands have been measured only tentatively, in preparing Fig. 2 we have assumed the values of I=9 and I=25/2$\hbar$, respectively. In calculations, the angular momenta I are identified with the average projections $\langle{I_y}\rangle$.

  

Figure 2: Experimental [9,28,11,10] and calculated alignments of the SD bands in ⁵⁹Cu (diamonds), ⁶⁰Zn (squares), and ⁶¹Zn (circles), relative to the SD band in ⁵⁸Cu (triangles). Calculations have been performed within the HF method with the SLy4 Skyrme interaction. In order to provide for a suitable reference at high rotational frequencies, two artificial gamma rays (open triangles) have been added to the ⁵⁸Cu experimental data (see text).

In Fig. 3 we present a similar comparison between the measured and calculated dynamic moments of inertia ${\cal J}^{(2)}$= $\partial{I(\omega)}/\partial{\omega}$. In ⁵⁸Cu, ⁵⁹Cu, and ⁶¹Zn we obtain very good theoretical description of measured relative alignments and second moments. This gives us strong arguments in favor of the assigned configurations. However, unexpectedly, the SD band in the doubly-magic SD nucleus ⁶⁰Zn deviates strongly from the theoretical predictions. This has been tentatively interpreted as an effect of the simultaneous alignment of the g_9/2 neutrons and protons [9], or as a manifestation of the T=0 n-p correlations [10]. In the present paper we perform the first calculations based on these two assumptions.

  

Figure 3: Same as in Fig. 2 but for the second moments of inertia ${\cal J}^{(2)}$.