Symmetry conserving configuration mixing description of odd mass nuclei

We present a self-consistent theory for the description of the spectroscopic properties of odd nuclei, which includes exact blocking, particle-number and angular-momentum projection, and configuration mixing. In our theory the pairing correlations are treated in a variation-after-projection approach and the triaxial deformation parameters are explicitly considered as generator coordinates. The angular-momentum and particle-number symmetries are exactly recovered. The use of the effective finite-range density-dependent Gogny force in the calculations provides an added value to the theoretical results. We apply the theory to the textbook example of ${}^{25}\mathrm{Mg}$ and, although this nucleus has been thoroughly studied in the past, we still provide a novel view of nuclear phenomena taking place in this nucleus. We obtain an overall good agreement with the known experimental energies and transition probabilities without any additional parameter such as effective charges. In particular, we clearly identify six bands, two of which we interpret as collective $\gamma$ bands.

Figure 1

Single-particle levels of ${}^{25}\mathrm{Mg}$ for neutrons in the HFB approach. The thick dashed line represents the Fermi level. The Nilsson quantum numbers $[N,n_z,m_l,\mathrm{\Omega }]$ are indicated for the relevant orbitals.

Figure 2

Contour plots of the potential energy surfaces as a function of $(\beta ,\gamma )$ for positive parity. (a) stands for the PNVAP approach (no angular momentum projection), see Eq. (24). (b)–(f) correspond to the PNAMP approximation, see Eq. (26), for the angular momentum $I$ quoted in the insets. The solid black contour lines go from 1–10 MeV in steps of 1 MeV. The dashed white lines start at zero and increase by 0.1 MeV. The zero-energy contour is only present if the minimum is flat enough. The angle $\gamma$ is given in degrees. In each panel the energies are relative to the corresponding energy minimum.

Figure 3

The same as Fig. 2 but for negative parity.

Figure 4

The expectation value of the particle number operator calculated with the intrinsic wave function of Eq. (6) for (a) positive-parity and (b) negative-parity neutrons at each point of the $(\beta ,\gamma )$ plane. See the text for details.

Figure 5

The spectrum of ${}^{25}\mathrm{Mg}$ from theory (left) and experiment (right), Ref. [48].

Figure 6

Squared collective wave functions of the band heads of ${}^{25}\mathrm{Mg}$ in the $(\beta ,\gamma )$ plane. The spin and parity of the different states is given in the inset of each plot. In each plot the value of the outer contour corresponds to one-tenth of the maximum value shown in the corresponding palette. Each contour is incremented by this amount up to the maximum value. The angle $\gamma$ is given in degrees.

Figure 7

Transition probabilities of the two lowest bands. In (b) the experimental values [48] are shown and in (a) the theoretical ones. The numbers in blue (dark gray) color correspond to $B\left(E2\right)$ values, in ${\mathrm{e}}^2{fm}^4$, and those in red (light gray) to $B\left(M1\right)$ ones, in ${\mu }_N^2$.

Figure 8

Collective wave functions of the excited states of band III of ${}^{25}\mathrm{Mg}$ in the $(\beta ,\gamma )$ plane with their signs. The spin and parity of the different states is given in the inset of each plot. The angle $\gamma$ is given in degrees.

Figure 9

Branching ratios of the $1/2_2$ band, (a) theory and (b) experiment.

Figure 10

Branching ratios for the decay of band IV, in (a) the theory and in (b) the experimental data according to the adopted values of Ref. [48].