Beyond Mean-Field Calculations for Odd-Mass Nuclei

Beyond mean-field methods are very successful tools for the description of large-amplitude collective motion for even-even atomic nuclei. The state-of-the-art framework of these methods consists in a generator coordinate method based on angular-momentum and particle-number projected triaxially deformed Hartree-Fock-Bogoliubov (HFB) states. The extension of this scheme to odd-mass nuclei is a long-standing challenge. We present for the first time such an extension, where the generator coordinate space is built from self-consistently blocked one-quasiparticle HFB states. One of the key points for this success is that the same Skyrme interaction is used for the mean-field and the pairing channels, thus avoiding problems related to the violation of the Pauli principle. An application to ${}^{25}\mathrm{Mg}$ illustrates the power of our method, as agreement with experiment is obtained for the spectrum, electromagnetic moments, and transition strengths, for both positive and negative parity states and without the necessity for effective charges or effective moments. Although the effective interaction still requires improvement, our study opens the way to systematically describe odd-$A$ nuclei throughout the nuclear chart.

Figure 1

Energy surface of the false vacuum (a), surfaces of the lowest energy found at a given deformation among the several nonprojected 1qp states of positive (b) and negative (c) parity, and surfaces of the lowest energy found at a given deformation after projection on $N$, $Z$, and $J^{\pi }=5/2^{+}$ (d), $3/2^{+}$ (e), and $3/2^{-}$ (f) and $K$ mixing among the 1qp states. All energies are displayed as a function of $\beta$ and $\gamma$ calculated from the mass density distribution of the 1qp state as defined in Ref. [8]. The deformation energy is relative to the minimum of each surface, indicated by a black dot. The offset of the minimum of each surface relative to the one of false vacua is indicated in the upper corner of each sextant.

Figure 2

Comparison of the calculated low-lying levels grouped into rotational bands with data taken from Ref. [22].

Figure 3

Excitation energies of states in the ground-state band of ${}^{25}\mathrm{Mg}$, and $B(E2)$ and $B(M1)$ values for transitions between them.