Shell structure of neutron-rich $Z$$\approx$20 nuclei

In a series of recent experiments performed at the Argonne National Laboratory with Gammasphere[11] and National Superconducting Cyclotron Laboratory,[12] properties of low-lying collective states of even-even neutron-rich titanium isotopes have been measured. As illustrated in Fig. 1, the data reveal the presence of a closed $N$=32 subshell, in addition to the standard $N$=28 shell present in heavier elements. Indeed, both $^{50}$Ti and $^{54}$Ti show the increased 2$^+$ energies, and decreased BE2 values, as compared to their neighbours.

Figure 1: Energies of the first excited 2$^+$ states (squares, left scale) and reduced transition probabilities BE2(0$^+$$\rightarrow$2$^+$) (circles, right scale), measured in neutron-rich Ti isotopes.[11,12]

In order to explain such a change of the shell structure, the single-particle neutron $\nu$f$_{5/2}$ orbital must be shifted up, which leaves a gap between the spin-orbit-split $\nu$p$_{3/2}$ and $\nu$p$_{1/2}$ orbitals, and creates a subshell closure for four particles occupying $\nu$p$_{3/2}$. The shell-model calculations,[15] performed for the single-particle orbitals shifted in this way, confirm the pattern shown in Fig. 1. The origins of the shift are attributed to the decreased monopole interaction energy between the proton $\pi$f$_{7/2}$ and neutron $\nu$f$_{5/2}$ orbitals, which occurs when protons are removed from $\pi$f$_{7/2}$. The source of such a monopole interaction is in turn attributed to the shell-model tensor interaction between these orbitals.

Positions of single-particle levels can be best studied within the mean-field approximation, in which they are basic dynamic characteristics of the system, resulting from the two-body interactions being averaged with particle densities of occupied states. Therefore, in this paper I evaluate the single-particle energies by applying the mean-field methods to tensor interactions.