Rotation of Neutron-Rich Ne and Mg Isotopes

To obtain a quantitative understanding of measured quadrupole moments, we performed systematic cranking calculations without pairing using the self-consistent cranked Skyrme Hartree-Fock (HF) method (code HFODD [52,53]) with the Skyrme parametrization SLy4 [54,55]. This method has shown to provide an accurate description of various properties of rotational bands in different mass regions (see, e.g., Refs. [56,57,58]). For the details pertaining to theoretical calculations, see forthcoming Ref. [29]. Here we only mention that the intrinsic configurations are labeled by means of total occupation numbers in each parity-signature sector $N_{\pi, r}$: [N_+,-i, N_+,+i, N_-,-i, N_+,+i]. For instance, the ground-state configuration of ³⁰Ne (two protons in the d_5/2 orbital; the neutron sd shell completely filled) can be written as [2233]_p[7733]_n. As seen in Fig. 2, the deformed intruder configurations in ³⁰Ne and ³²Mg can be associated with 2-particle, 2-hole neutron excitation to the f_7/2shell; hence it can be written as [6644]_n. The corresponding neutron single-particle Routhian diagram for ³⁰Ne is shown in Fig. 6.

  

Figure 6: Neutron single-particle Routhians in ³⁰Ne calculated in the HF+SLy4 model for the [2233]_p [6644]_n intruder configuration.

Calculations of rotational bands were carried out for the deformed configurations in ^{30,32,34,36,38}Ne and ^{32,34,36,38,40}Mg. Since pairing correlations were ignored, we consider these calculations as exploratory. Here, our main objective is to investigate the effect of fast rotation on properties of weakly bound systems with a very large neutron excess. For the deformed bands considered, the proton configurations for Ne and Mg are always [2233]_p and [3333]_p, respectively. (There are no crossings between positive and negative-parity Routhians for Z=10 and 12 in the range of angular momentum discussed.) As far as the neutrons configurations are concerned, we assumed [6644]_n for N=20, [6655]_n and [7744]_n for N=22, [7755]_n for N=24, [7766]_n for N=26, and [7777]_n for N=28. These configurations correspond to the lowest deformed bands in the neutron-rich Ne and Mg nuclei investigated.

  

Figure 7: Neutron (gray arrows) and proton (black arrows) quadrupole deformations $\beta_2$ and $\gamma$ as functions of rotational frequency for the deformed bands in ^{30,32,34,36,38}Ne and ^{32,34,36,38,40}Mg.

From the calculated components of the quadrupole moment, Q₂₀ and Q₂₂, one can extract the Bohr quadrupole deformation parameters $\beta_2$and $\gamma$ [53]:

$$\tan{\gamma}={Q_{22} \over Q_{20}},~~\beta_2=\sqrt{{\pi\over 5}} {\sqrt{Q_{20}^2+Q_{22}^2 }\over {N_\tau \langle r^2\rangle}},$$ (1)

where $\tau$=1 (-1) for neutrons (protons), and N₁=N and N_-1=Z. Figure 7 shows the calculated proton and neutron deformation trajectories in the $(\beta_2,\gamma)$ plane as functions of rotational frequency. In general, the differences between proton and neutron deformations are rather small. They are most pronounced at low spins; the proton and neutron shapes become similar at angular momenta close to the termination limit. The strongest isovector effects are predicted in the N=26 isotones where $\Delta\beta_2$$\equiv$ $\beta^p_2-\beta^n_2$ reaches 0.05 at low spins. It is interesting to see that for ³⁶Ne $\Delta\beta_2$ changes sign from postive ( $\beta^p_2>\beta^n_2$) at low spins to negative ( $\beta^p_2<\beta^n_2$) at very high angular momenta. The pattern represented in Fig. 7 is characteristic of rotational structures built upon relatively few valence nucleons. The loss of collectivity is rather fast due to the alignment of f_7/2 neutrons, which leads to the band termination at relatively low spins (e.g., I=12 for ³⁰Ne).

An interesting question which is often asked in the context of rotational motion of weakly bound neutron-rich nuclei is whether the weakly bound neutrons could be kicked off the nucleus due to the large centrifugal force. To shed some light on this problem, Fig. 8 displays the predicted rms proton and neutron radii as functions of $\omega$. Generally, rms radii very weakly depend on rotation. The small reduction calculated in some cases comes primarily from the decrease in $\beta_2$ along the band termination path. However, the deformation effect is weaker compared to the bulk dependence of radii on Z and N.

  

Figure 8: Neutron and proton rms radii as a function of rotational frequency for the deformed bands in ^{30,32,34,36,38}Ne and ^{32,34,36,38,40}Mg.

Considering the results presented in Figs. 7 and 8, one can conclude that the isovestor effects are not very pronounced at high angular momenta in the neutron-rich Ne and Mg isotopes. This is not entirely unexpected. In these nuclei, the valence neutrons occupy f_7/2 high-j intruder states which, due to their large orbital angular momentum (i.e., large centrifugal barrier), are fairly well localized within the nuclear volume in spite of their weak binding. One can say that in most cases, as a result of the Coriolis force, the low-$\ell$ states (which are natural candidates for halo effects) are going to be crossed at high rotational frequencies by the high-$\ell$ intruder orbitals. Consequently, the tendency to develop a halo should be reduced at high spins. The full analysis of our calculations of the Ne and Mg isotopes, containing the discussion of the moments of inertia, will be presented in a forthcoming paper [29]. The general conclusion is that the response of very neutron-rich nuclei to rotation is fairly ``normal"; no decoupling of the valence (skin) neutrons at high spins is predicted.