Deformation-induced splitting of the isoscalar $E0$ giant resonance: Skyrme random-phase-approximation analysis

The deformation-induced splitting of isoscalar giant monopole resonance (ISGMR) is systematically analyzed in a wide range of masses covering medium, rare-earth, actinide, and superheavy axial deformed nuclei. The study is performed within the fully self-consistent quasiparticle random-phase-approximation method based on the Skyrme functional. Two Skyrme forces, one with a large (SV-bas) and one with a small (SkP) nuclear incompressibility, are considered. The calculations confirm earlier results that, because of the deformation-induced $E0-E2$ coupling, the isoscalar $E0$ resonance attains a double-peak structure and significant energy upshift. Our results are compared with available analytic estimations. Unlike earlier studies, we get a smaller energy difference between the lower and upper peaks and thus a stronger $E0-E2$ coupling. This in turn results in more pumping of $E0$ strength into the lower peak and more pronounced splitting of ISGMR. We also discuss widths of the peaks and their negligible correlation with deformation.

Figure 1

ISGMR (top panels) and ISGQR($K$ = 0) (bottom panels) strength functions in ${}^{142,146,150}\mathrm{Nd}$, calculated within QRPA. The calculated deformation parameters are indicated for each isotope. The experimental data (energy centroids and widths) [13] are shown by arrows and horizontal bars.

Figure 2

Isoscalar $E0$ strength functions in deformed ${}^{154}\mathrm{Sm}$ calculated within SRPA (red dotted lines) and QRPA (black solid lines) with the forces SV-bas (left) and SkP (right). The SRPA calculations are performed without (top panels) and with (bottom panels) the $E0-E2$ coupling. For the comparison, the $E0$ data from RCNP [14] (blue filled squares) and TAMU [6] (green filled circles) experiments are depicted. In bottom panels, the calculated peak energies of the $\lambda \mu$ = 20 branch of ISGQR are marked by arrows.

Figure 3

Isoscalar $E0$ strength functions in Cd isotopes, calculated within QRPA with the forces SV-bas (black solid line) and SkP (red dotted line). For the comparison, the $E0$ data from RCNP [16] (blue filled squares) and TAMU [17] (green filled circles, only for ${}^{116}\mathrm{Cd}$) experiments are depicted. The calculated peak energies of the quadrupole branch with $K$ = 0 are marked by arrows. The experimental deformation parameters $\beta$ [29] are indicated for each isotope.

Figure 4

The QRPA isoscalar $E0$ strength functions in deformed ${}^{164}\mathrm{Dy}, {}^{168}\mathrm{Er}, {}^{172}\mathrm{Yb}$, and ${}^{238}\mathrm{U}$, calculated with (black solid line) and without (red dotted line) the equilibrium deformation (indicated in the plots). The force SV-bas is used. The peak energies of ISGQR($K$ = 0) branch are marked by arrows.

Figure 5

The QRPA isoscalar $E0$ strength functions for isotopes ${}^{242,254,270}\mathrm{Nd}$ (left) and ${}^{264,284,304}114$ (right) calculated at equilibrium deformations (indicated in the figure). The force SV-bas is used. The peak energies of ISGQR($K$ = 0) branch are marked by arrows.

Figure 6

Dependence of the calculated ISGMR energies (black filled symbols) on the mass number $A$. For ${}^{106-116}\mathrm{Cd}, {}^{142,146,148}\mathrm{Nd}$, and ${}^{154}\mathrm{Sm}$, the experimental values (red open symbols) are shown. The red line gives the estimate $E_{\mathrm{M}}^0\approx 80A^{-1/3}$ MeV.

Figure 7

(a) Dependence of the calculated deformation shift $d{E}_{\mathrm{M}}$ on the mass number $A$. (b) Dependence of $d{E}_{\mathrm{M}}{A}^{1/3}$ on the squared deformation parameter ${\beta }^2$. SS (red solid lines) and VM (blue dotted line) estimates are depicted.

Figure 8

(a) Dependence of the calculated deformation splitting $\mathrm{\Delta }{E}_{\mathrm{M}}$ (black filled symbols) on the deformation parameter $\beta$. (b) The same but for $\mathrm{\Delta }{E}_{\mathrm{M}}{A}^{1/3}$. SS estimates (5a)–(5c) and VM estimates (11) are depicted by red solid and blue dotted lines, respectively. For SS, the estimates with (upper line) and without (lower line) $E0$–$E2$ coupling are shown. See more details in the text.

Figure 9

Calculated fractions $S_1$ and $S_2$ (in $\%$) of $E0$ strength in low- and high-energy branches of ISGMR in well-deformed nuclei. The dependence on the mass number (a) and deformation (b) is exhibited.

Figure 10

Calculated widths $w_1$ and $w_1$ of lower and upper ISGMR branches in well-deformed nuclei. The dependence on the mass number (a) and deformation (b) is exhibited.