Systematics of the First $2^{+}$ Excitation with the Gogny Interaction

We report the first comprehensive calculations of $2^{+}$ excitations with a microscopic theory applicable to over 90% of the known nuclei. The theory uses a quantal collective Hamiltonian in five dimensions. The only parameters in theory are those of the finite-range, density-dependent Gogny D1S interaction. The following properties of the lowest $2^{+}$ excitations are calculated: excitation energy, reduced transition probability, and spectroscopic quadrupole moment. We find that the theory is very reliable to classify the nuclei by shape. For deformed nuclei, average excitation energies and transition quadrupole moments are within 5% of the experimental values, and the dispersion about the averages are roughly 20% and 10%, respectively. Including all nuclei in the performance evaluation, the average transition quadrupole moment is 11% too high and the average energy is 13% too high.

Figure 1

Scatter plot of 519 even-even nuclei as a function of their experimental and theoretical $2^{+}$ excitation energies.

Figure 2

Histogram of the logarithmic errors $R_E,R_Q$ of the present theory. Left-hand panel: Energies of the first excited $2^{+}$ states for 519 of the 557 nuclei whose excitation energies are tabulated in Ref. 10. Right-hand panel: Matrix elements $⟨2||{\widehat{Q}}_2||0⟩$ for the transition between the ground and first excited state for 318 of the 328 nuclei whose $B(E2)$ values are tabulated in Ref. 10.

Figure 3

Experiment compared to theory for the $B(E2,0^{+}\to 2^{+})$ for the nuclei tabulated in Ref. 10. This graph may be directly compared with their Fig. C. Values within the lines are within a factor of 2 of experiment. Of the 306 cases shown here, 93% are within the error band. This is superior to their “global” phenomenological fit and is much better than the theoretical models they consider.

Figure 4

Experiment compared to theory for the quadrupole moment of 98 excited $2^{+}$ states. Experimental database is from the tabulation in Ref. 12.