Level crossings and the HF convergence

In all calculations, we diabatically follow configurations, i.e., we always occupy orbitals with given single-particle characteristics, irrespectively of whether they cross with the other orbitals or not. This is technically easy if crossings involve orbitals of different parity/signature blocks, but special techniques [36] must be used in self-consistent methods, to diabatically follow configurations which cross within the common parity/signature blocks. Those crossings are particularly interesting because they usually give rise to an up-bending or back-bending structures, and are thus important for the experimental identification of the underlying structures.

In the present study we separate the diabatic configurations by proceeding as follows. If, in the positive-parity orbitals, a particle ``switches on and off'' from the occupation of one orbital to another, we force an occupation of the state that has a larger single-particle alignment, independently of whether it is slightly higher or lower in energy. In the present case this implies that we always force the particle into the [211]1/2 orbital and leave the extruder orbital [202]5/2 empty. Incidentally, by occupying the state that has a smaller alignment, at the end of a successful iteration sequence we obtain a markedly different solution, with much smaller deformations, i.e., the fact that two configurations mix does not imply that both manifest all the same physical properties in this case.

A rich collection of the experimental data on the back-bending and up-bending phenomena that exist today in the literature has been interpreted in terms of the theoretical single-particle configurations followed according to the diabatic scheme. Whether the experimental bands exist that follow an adiabatic scheme is an open question, and an unambiguous (while anyway model depended) demonstrations are very difficult. On the theory grounds, this question cannot be settled within a mean-field approach. Therefore, our approach to follow diabatic configurations is dictated by the fact that a great majority of the high spin data has been interpreted accordingly. In case of need demonstrated by future experiments, our present results could immediately be used as a first step in the band-band mixing calculations.

By examining the routhian diagrams obtained self-consistently at a fixed particle-hole configuration, as, e.g., those shown in Figs. 1 and 2, one cannot predict crossings which may happen in some other configuration. This is especially true in ³²S, where different configurations correspond to fairly different deformations, see Sec. 4.2, and therefore, may involve significantly different ordering of orbitals. As an example, in Figs. 4 and 5 we present neutron routhians corresponding to the self-consistently calculated 3¹_-3¹_- and 3¹₊3¹₊bands, respectively. Since these configurations produce deformations significantly smaller than that of the magic SD configuration, the (empty) extruder orbitals [202]5/2(r=$\pm {i}$) are here much lower in energy, and strongly mix with the (occupied) [211]1/2(r=$\pm {i}$) orbitals. In these configurations, the signature splitting is very large, and strongly depends on the actual configuration, see discussion in Sec. 4.4. Therefore the order of routhians in the 3¹₊3¹₊ configuration (Fig. 5) is entirely different from that in the 3¹_-3¹_- configuration (Fig. 4) and leads to very strong mixing and level repulsion at and near the crossing frequency in the latter case.

We can easily identify the crossing regions by the simple fact that the HF iterations are poorly convergent, or non-convergent there [36]. This concerns only those methods of solving the HF equations, which are based on successive diagonalizations of the mean-field Hamiltonian. The gradient method and the imaginary-time method [39], always arrive at the smallest-energy solution, and do converge. However, the obtained solutions simply correspond to one of the infinitely many possible mixed-orbital solutions, with the same or very close energy, and consequently, those methods do not cure the problem, but allow for not seeing it.

We have made every possible effort to achieve convergence of all configurations at all angular frequencies, however, in several cases it turned out not to be possible. This is the case, for example, for the 3¹₊3¹₊ band at $\hbar\omega$=1.0-1.4MeV; the non-convergence here results in an irregular behavior of routhians in Fig. 2. In the following, we show energies corresponding to the non-converged points along with the well-converged ones, however, we remove points corresponding to non-converged solutions from plots of other observables.