Covariant energy density functionals

The RHB equations [52,8] were solved in the basis of an anisotropic three-dimensional harmonic oscillator in Cartesian coordinates. For all nuclei and states determined in this work, the same basis deformation of $\beta_0=0.3$, $\gamma=0^{\circ}$ and oscillator frequency of $\hbar \omega_0=41$A$^{-1/3}$MeV have been used. All fermionic and bosonic states belonging to the shells up to $N_F=14$ and $N_B=20$ were taken into account when performing diagonalization of the Dirac equation and matrix inversion of the Klein-Gordon equations, respectively. As follows from detailed analysis of Refs. [8,13], this truncation of basis provides sufficient accuracy of the calculations.

As the effective interaction in the particle-particle ($pp$) channel, the central part of the non-relativistic Gogny finite-range interaction was used. The clear advantage of such a pairing force is that it provides an automatic cutoff of high-momentum components. The motivation for such an approach to the description of pairing was given in Ref. [52]. The D1S parametrization of the Gogny force was used here. No specific adjustment of its strength was used, because it provided a reasonable description of the pairing indicators in $^{249}$Bk and $^{249,251}$Cf and moments of inertia in $^{252,254}$No [8].

Two different covariant EDFs, namely, NL1 [23] - fitted to the nuclei in the valley of beta-stability, and NL3* [24] - tailored towards the description of neutron-rich nuclei, were used in the current study. The covariant EDF NL1 was extensively used in the calculations of rotational bands across the nuclear chart (see Ref. [20]). Covariant EDF NL3* was less tested than NL1 with respect to the description of rotating nuclei. However, its global performance is well established [53]. Note that so far only these two covariant EDFs were systematically confronted with experimental data on single-particle states. For example, the study of predominantly single-particle states in odd-mass nuclei neighbouring to the doubly magic spherical nuclei was performed in Ref. [54] within the relativistic particle-vibration coupling model employing covariant EDF NL3*. In Ref. [13], the first systematic study of the single-particle spectra in deformed nuclei in rare-earth region and actinides was performed with the covariant EDFs NL1 and NL3*.

It is interesting that the overall accuracy of the description of the energies of deformed one-quasiparticle states [13] is slightly better in the old covariant EDF NL1 than in the recent functional NL3*. This suggests that the inclusion of extra information on neutron rich nuclei into the fit of the functional NL3* may lead to some degradation of the description of single-particle states along the valley of beta-stability. Note that these two functionals well reproduce deformation properties of ground states of even-even actinides [8,20] and indicate that they are axially symmetric.

A proper description of odd or rotating nuclei implies breaking of the time-reversal symmetry of the mean field, which is induced by the unpaired nucleon [55] or rotation [56]. As a consequence, time-odd mean fields and nucleonic currents, which cause the nuclear magnetism [57] have to be taken into account. In the covariant EDF, time-odd mean fields are defined through the Lorentz invariance, and thus they do not require additional coupling constants.

The effects of blocking due to the odd particle were included in a fully self-consistent way. This was done within the code CRHB, according to Refs. [58,59,60]. The blocked orbital was specified by different tags such as (i) dominant main oscillator quantum number $N$ of the wave function, (ii) dominant $\Omega$ quantum number of the wave function, (iii) particle or hole nature of the blocked orbital, and (iv) position of the state within the specific parity/signature/dominant-$N$/dominant-$\Omega$. For a given odd-mass nucleus, possible blocked configurations were defined from the analysis of calculated quasiparticle spectra in neighboring even-even nuclei and the occupation probabilities of the single-particle orbitals of interest in these nuclei.

Note that in the cases when the calculations of odd-mass nuclei were performed only for the definition of the $\Delta^{(3)}$ indicators (see Sec. 3.4 below), we restricted the analysis to 5-6 one-quasiparticle configurations with expected lowest total energies, so as to properly determine the ground state of an odd-mass nucleus. The calculations confirmed the conclusion of the statistical analysis of Ref. [13] that absolute majority of the one-quasiparticle configurations are axially symmetric. However, some degree of triaxiality was obtained in the $\nu 3/2[622]$, $\nu 1/2[501]$ and $\pi 1/2[400]$ configurations.