Nuclear spectra from low-energy interactions

A method to describe spectra starting from nuclear density functionals is explored. The idea is based on postulating an effective Hamiltonian that reproduces the stiffness associated with collective modes. The method defines a simple form of such an effective Hamiltonian and a mapping to go from a density functional to the corresponding Hamiltonian. In order to test the method, the Hamiltonian is constrained using a Skyrme functional and solved with the generator-coordinate method to describe low-lying levels and electromagnetic transitions in ${}^{48,49,50,52}\mathrm{Cr}$ and ${}^{24}\mathrm{Mg}$.

Figure 1

HF energy versus deformation $\beta$ for ${}^{48,50,52}\mathrm{Cr}$ using a basis of 12 spherical oscillator shells ($N_{\max }=11$). Experimental binding energies are shown with dashed lines. The SLy4 EDF results are obtained with the code [25] and the results abbreviated SLy4-H are obtained with the effective Hamiltonian (1).

Figure 2

Calculated positive parity and even spin spectra for ${}^{48}\mathrm{Cr}$, (a) without temperature excitations and in (b) with inclusion of single-particle type excitations using the temperature method (16) with $b=0.45$. In both panels we used the SLy4-H Hamiltonian with 11 oscillator shells ($N_{\max }=10$) for the single-particle basis and a grid consisting of $N_{\varphi }=191$ HFB vacua for the many-body basis. Experimental data taken from Ref. [32]. The Hamiltonian for each spin has been diagonalized using the number of natural states found from the Hill-Wheeler equation (28) considering the yrast state.

Figure 3

HFB energy versus deformation for ${}^{48}\mathrm{Cr}$ with the SLy4-H Hamiltonian. Twelve oscillator shells are used for the single-particle basis ($N_{\max }=11$), $j_x=0$ and the pairing interaction strength is kept fixed $(g_p,g_n)=(1,1)$. The $(\beta ,\gamma )$ plane is drawn with the standard Lund convention with $\gamma ={60}^{\circ }$ along the positive $y$ axis.

Figure 4

Positive parity and even spin spectra including $B\left(E2\right)$ transitions for ${}^{48}\mathrm{Cr}$. Theory to the left and experiment to the right. The experimental values for the yrast band are also shown together with theory for easier comparison. The strengths for the transitions are indicated by the thickness of the lines and are explicitly given in ${\mathrm{e}}^2{\mathrm{fm}}^4$ for the yrast band.

Figure 5

$B\left(E2\right)$ transitions ($I\to I-2$) and spectroscopic quadrupole moments for the yrast band in ${}^{48}\mathrm{Cr}$. No experimental data has been found for $Q_{\mathrm{spec}}$.

Figure 6

Released $\gamma$-ray energies for transitions $I+2\to I$ for the yrast band of ${}^{48}\mathrm{Cr}$. The back bending at $I=10\hslash$ indicates a change in the internal structure.

Figure 7

Energies of the GS rotational band in ${}^{49}\mathrm{Cr}$ computed in different bases. In (a) the basis includes 116 states and the cranking is switched off. In (b) and (c), respectively, 114 and 156 states are included in the basis and the cranking is switched on. Results on the left (right) side of (b) and (c) are obtained in a basis with signature $r_x=i$ ($-i$).

Figure 8

Excitation energies of states in the GS rotational band in ${}^{49}\mathrm{Cr}$. The black squares denote experimental data, whereas the red circles show the results of the computation in the largest basis considered (156 states) with cranking. For each $I$ the computed energies corresponds to the lowest energies among the spectra displayed in Fig. 7. The dash and dotted lines denotes the two branches, which connect the calculated states with $\mathrm{\Delta }I=2$ (see text).

Figure 9

Same as Fig. 4 for the spectra and $B\left(E2\right)$ transitions in ${}^{50}\mathrm{Cr}$.

Figure 10

$B\left(E2\right)$ transitions and spectroscopic quadrupole moments for the yrast band in ${}^{50}\mathrm{Cr}$.

Figure 11

Same as Fig. 4 for the spectra and $B\left(E2\right)$ transitions in ${}^{52}\mathrm{Cr}$. The two identified experimental bands are indicated with thin lines when there are no measured $B\left(E2\right)$ values.

Figure 12

Transitions and quadrupole moments for ${}^{52}\mathrm{Cr}$. The dashed lines are for the second band.

Figure 13

Same as Fig. 4 for spectra and transitions in ${}^{24}\mathrm{Mg}$. The experimental values for the $I=8$ states, both energies and the transitions, are taken from Ref. [55].

Figure 14

Transitions and quadrupole moments for ${}^{24}\mathrm{Mg}$.

Figure 15

Calculated low-energy spectra for ${}^{24}\mathrm{Mg}$ as a function of the parameter $b$ (19). The energies $E$ are relative to the ground-state energy for $b=0.45$. The states shown in the plot are the yrast states (filled lines) and the first two excited states (dashed lines) that have excitation energy below 9 MeV. The used values for $b$ are distributed between 0.01 and 0.95.