Systematics of the first $2^{+}$ excitation in spherical nuclei with the Skryme quasiparticle random-phase approximation

We use the quasiparticle random-phase approximation (QRPA) and the Skyrme interactions $\mathrm{SLy}4$ and ${\mathrm{SkM}}^{*}$ to systematically calculate energies and transition strengths for the lowest $2^{+}$ state in spherical even-even nuclei. The ${\mathrm{SkM}}^{*}$ functional, applied to 178 spherical nuclei between $Z=10$ and 90, produces excitation energies that are on average 11% higher than experimental values, with residuals that fluctuate about the average by $-35\%$ to $+55\%$. The predictions of ${\mathrm{SkM}}^{*}$ and $\mathrm{SLy}4$ have significant differences, in part because of differences in the calculated ground state deformations; ${\mathrm{SkM}}^{*}$ performs better in both the average and dispersion of energies. Comparing the QRPA results with those of generator-coordinate-method (GCM) calculations, we find that the QRPA reproduces trends near closed shells better than the GCM, and that it overpredicts the energies less severely in general.

Figure 1

Particle-hole character of the lowest $2^{+}$ solutions. The histogram displays the quantity $\Delta N$ defined in Eq. (1) for 155 nuclei in the $\mathrm{SLy}4$ data set (one of which we drop—see text). The values $-2,0,+2$ correspond to excitations of hole-hole, particle-hole, and particle-particle character, respectively.

Figure 2

Distribution of the summed $(X+Y){}^2$ amplitudes, Eq. (2), for the first $2^{+}$ transitions in spherical nuclei, for the $\mathrm{SLy}4$ functional. Values greater than 4 are consolidated in the last bin.

Figure 3

Calculated energies for lowest $2^{+}$ states in spherical nuclei, plotted vs experimental energies. The theoretical energies are from QRPA with the $\mathrm{SLy}4$ energy functional. The experimental data are from Ref. [14]. For triangles, see text.

Figure 4

Histogram of the quantity $R_E$ for the 155 nuclei in the $\mathrm{SLy}4$ data set. The highest bin includes three nuclei having $R_E>2$.

Figure 5

Same as Fig. 3, but using the ${\mathrm{SkM}}^{*}$ functional for the calculated energies.

Figure 6

$2^{+}$ excitation energies in the vicinity of doubly magic nuclei. Displayed on either side of the doubly magic nucleus $(N,Z)$ are the adjacent even-even isotopes $(N-2,Z)$ and $(N+2,Z)$. Theoretical energies (open circles) were calculated with the ${\mathrm{SkM}}^{*}$ functional. Filled circles are experimental energies.

Figure 7

Same as Fig. 6, but for isotones $(N,Z-2),(N,Z+2)$ adjacent to doubly magic nuclei $(N,Z)$.

Figure 8

Same as Fig. 3, but for $B(E2)\uparrow$.

Figure 9

Histogram of $R_Q$ for the 155 nuclei in the $\mathrm{SLy}4$ data set. The highest bin represents the nucleus ${}^{210}\mathrm{Po}$.

Figure 10

Lowest $2^{+}$ energies of even Ni, Sn, and Pb isotopes [panels (a) and (b)], and isotones with $N=$ 28, 50, and 82 [panel (c)]. The open circles represent our results (${\mathrm{SkM}}^{*}$), the squares the GCM-5DCH results of Ref. [4], and the filled circles experimental data.