Discussion of the energy density at N 3 LO with conserved spherical, space-inversion, and time-reversal symmetries (Sec. VI in Ref. [1]) was left unfinished in the sense that properties of secondary densities were not taken into account. By this omission, within these symmetry conditions, five extra independent terms appeared in the energy density functional (EDF).The problem was related to the fact that the spherical symmetry imposes specific relations between derivatives of vector or tensor fields. In particular, following Eq. (44), an arbitrary vector ( J) or tensor ( ↔ R) field must have the form dictated by the generalized Cayley-Hamilton theorem [2]:where J (r) and R(r) are scalar functions, that is, functions of r = |r|. Then it is a matter of simple algebra to show thatwhere primes denote derivatives with respect to r. The first two equalities constitute relations between pairs of secondary densities, which are imposed by the spherical symmetry. Therefore, the corresponding terms in the energy densities at fourth and sixth orders (Eqs. (61) and (62) in Ref.[1]) are pairwise identical. Specifically, this concerns pairs of terms represented by the coupling constants C 1 21 and D 1 21 in fourth order andConsequently, the numbers of terms given in Table XX in Ref.[1] at fourth and sixth orders are smaller by one and four, respectively, and are given in the second column in Table I here.In addition, we correct here two misprints found in Ref. [1]. First, the description of rules that were used to select terms in the functional, which was given before Eq. (31), should read as follows: To avoid double-counting one takes only terms with

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We derive a zero-range pseudopotential that includes all possible terms up to sixth order in derivatives. Within the Hartree-Fock approximation, it gives the average energy that corresponds to a quasi-local nuclear Energy Density Functional (EDF) built of derivatives of the one-body density matrix up to sixth order. The direct reference of the EDF to the pseudopotential acts as a constraint that divides the number of independent coupling constants of the EDF by two. This allows, e.g., for expressing the isovector part of the functional in terms of the isoscalar part, or vice versa. We also derive the analogous set of constraints for the coupling constants of the EDF that is restricted by spherical, space-inversion, and time-reversal symmetries.

Spectroscopy of Element 115 Decay Chains

Important insight into the symmetry properties of the nuclear ground-state (gs) shape is obtained from the characteristics of low-lying collective energy-level spectra. In the 1950s, experimental and theoretical studies showed that in the gs many nuclei are spheroidal in shape rather than spherical. Later, a hexadecapole component of the gs shape was identified. In the 1970-1995 time frame, a consensus that reflection symmetry of the gs shape was broken for some nuclei emerged. Here we present the first calculation across the nuclear chart of axial symmetry breaking in the nuclear gs. We show that we fulfill a necessary condition: Where we calculate axial symmetry breaking, characteristic gamma bands are observed experimentally. Moreover, we find that, for those nuclei where axial asymmetry is found, a systematic deviation between calculated and measured masses is removed.

Products of the fusion-evaporation reaction 48 Ca + 243 Am were studied with the TASISpec set-up at the gas-filled separator TASCA at the GSI Helmholtzzentrum für Schwerionenforschung. Amongst the detected thirty correlated α-decay chains associated with the production of element Z = 115, two recoil-α-fission and five recoil-α-α-fission events were observed. The latter are similar to four such events reported from experiments performed at the Dubna gas-filled separator. Contrary to their interpretation, we propose an alternative view, namely to assign eight of these eleven decay chains of recoil-α(-α)-fission type to start from the 3n-evaporation channel 288 115. The other three decay chains remain viable candidates for the 2n-evaporation channel 289 115.