Skyrme energy density functional SLy4

For the Skyrme EDF SLy4 [26], the calculations were carried out with the Skyrme HFB solver CR8 whose development over the years has been documented in Refs. [31,32,33,34]. It uses a 3D coordinate space mesh representation of single-particle states along the lines of the solver EV8 described in Ref. [35], but is extended in such a way that intrinsic time-reversal invariance can be broken and that HFB equations are solved instead of HF+BCS. Single-particle states are represented in a cubic box of $32^3$fm$^3$ with a step size of 0.8fm between discretization points. Imposing triaxial symmetry, only 1/8 of the box has to be represented numerically, meaning that only a $20 \times 20 \times 20$ mesh is to be treated.

When calculated with SLy4, the ground states of even-even nuclei considered here are all axial, and the blocked states of odd-$A$ nuclei also remain almost axial. All blocked calculations were initialized with the ground states of adjacent even-even nuclei. Self-consistent blocking was performed by considering the quasiparticle state dominated by a given eigenstate of the single-particle Hamiltonian and by exchanging the corresponding columns of the HFB $U$ and $V$ matrices after the diagonalization of the HFB Hamiltonian, which in turn was constructed using the mean fields of the blocked solution from the previous iteration, see, e.g., Refs. [32,36,37].

To avoid mixing of quasiparticle states with different average values of the angular momentum component $\langle \hat{j}_\parallel \rangle$ parallel to the symmetry axis of the initial configuration in the diagonalization of the HFB Hamiltonian, the many-body expectation value of $\langle \hat{J}_\parallel \rangle$ was held fixed with a cranking constraint at the value of $\langle \hat{J}_\parallel \rangle$ equal to the one of the blocked quasiparticle state. As the code CR8 allows for triaxial shapes the mixing cannot be fully suppressed. As a consequence, the blocked HFB states are not necessarily orthogonal even when they have different average value of $\langle \hat{J}_\parallel \rangle$.

In this respect, blocking the $\langle \hat{j}_\parallel \rangle = 1/2$ levels presents a particular difficulty. Without a cranking constraint, the code CR8 very often converges toward a solution where the many-body expectation value $\langle \hat{J}_\parallel \rangle$ is close to zero, and where the blocked quasiparticle in the spectrum of eigenstates of the HFB Hamiltonian is mixed with other low-lying quasiparticles of different $\langle \hat{j}_\parallel \rangle$. For these states, using or not using the cranking constraint might in some cases make a difference of the order of 100 to 200keV. An example is the ground state of $^{251}$Cf. In blocked calculations with cranking constraint, there is a low lying $\langle \hat{J}_\parallel \rangle^\pi = 1/2^+$ level at 45keV excitation energy above the calculated $\langle \hat{J}_\parallel \rangle^\pi = 3/2^+$ ground state. In the calculations without cranking constraint, the energy of the $3/2^+$ state does not change much, but the $1/2^+$ level is lowered by about 180keV and becomes the ground state, in agreement with experiment, but at the expense of the blocked quasiparticle being a strong mixture of $\langle \hat{j}_\parallel \rangle = 1/2$ and $\langle \hat{j}_\parallel \rangle = 3/2$, and of having an angular momentum $\langle \hat{J}_\parallel \rangle$ that cannot be easily interpreted within the strong coupling model anymore. For blocked states with higher $\langle \hat{j}_\parallel \rangle$, the mixing is always much smaller.

For the coupling constants of the so-called time-odd terms that contribute to cranked and blocked states, the same "hybrid" choice was made as in many earlier calculations [7,31,32,33,38,39,40,41,42,43,44]. They were set to the "native" values dictated by a density-dependent Skyrme two-body force for all terms except for those that multiply terms that couple two derivatives and two Pauli spin matrices. The latter were set to zero for reasons of Galilean invariance and internal consistency, cf. Refs. [31,38] for further discussion.

In the present study, neutron and proton surface pairing interactions were used, with the strengths adjusted to the three-point gaps $\Delta^{(3)}_n$ of $^{247}$ Cf ($N=149$) and $\Delta^{(3)}_p$ of $^{249}$ Bk ($Z=97$), as defined by Eqns. (7) and (6) below, leading to

$$V_{n} = -1240 \; \mbox{MeV} \; \mbox{fm}^{3},$$ (1)

$$V_{p} = -1575 \; \mbox{MeV} \; \mbox{fm}^{3}.$$ (2)

The pairing-active spaces were limited by the soft cutoffs of 5MeV above and below the neutron and proton Fermi energies as described in Ref. [45]. With that, the proton pairing strength is significantly larger than that given by the standard value of $V_n = V_p = -1250$MeVfm$^{3}$ used for heavy nuclei in previous publications [39,7,40,41,42,43] (which all have been carried out using the Lipkin-Nogami (LN) scheme, though). These previous values were obtained by adjusting moments of inertia in rotational bands of heavy nuclei [45]. There is, however, also some published work on very heavy nuclei [44,46] where the pairing interaction of volume type was adjusted to some $\Delta^{(3)}_q$ values of odd-$A$ actinide nuclei.

In principle, the like-particle pairing interaction should be of isovector type, which implies $V_{n} = V_{p}$. Differences between the adjusted neutron and proton pairing strengths can have many reasons: (i) the compensation of imperfections of the calculated single-particle spectra, (ii) the compensations of the cutoff energy that is chosen to be the same in neutron and proton phase spaces (iii) the compensation of the imperfections of the chosen form of the pairing interaction itself. Note that if there were none of the above mentioned deficiencies, the proton pairing strength would have to be smaller than the neutron pairing strength, so as to compensate for the absence of Coulomb pairing in our calculations.

Our attempts to adjust the neutron pairing strength to $\Delta^{(3)}_n$ in $^{251}$ Cf ($N=153$), as done in Ref. [21] and in the present work for UNEDF2, led to the values of $V_{n} = -1060$ and $V_{p} = -1565$MeVfm$^{3}$. With such much weaker neutron pairing strength, in ground states of many of the lighter odd-$N$ nuclei, the pairing disappeared. This seems to be connected to an anomaly of the calculated single-particle spectra of heavy $N=153$ isotones that translates into a much larger values of $\Delta^{(3)}_n$ for these nuclei than for the neighbouring ones.