Isoscalar giant resonances in the Sn nuclei and implications for the asymmetry term in the nuclear-matter incompressibility

We have investigated the isoscalar giant resonances in the Sn isotopes using inelastic scattering of $386$-MeV $\alpha$ particles at extremely forward angles, including $0^{°}$. We have obtained completely “background-free” inelastic-scattering spectra for the Sn isotopes over the angular range $0^{°}$–$9^{°}$ and up to an excitation energy of $31.5$ MeV. The strength distributions for various multipoles were extracted by a multipole decomposition analysis based on the expected angular distributions of the respective multipoles. We find that the centroid energies of the isoscalar giant monopole resonance (ISGMR) in the Sn isotopes are significantly lower than the theoretical predictions. In addition, based on the ISGMR results, a value of $K_{\tau }=-550\pm 100$ MeV is obtained for the asymmetry term in the nuclear incompressibility. Constraints on interactions employed in nuclear structure calculations are discussed on the basis of the experimentally obtained values for $K_{\infty }$ and $K_{\tau }$.

Figure 1

Vertical-position spectrum at the focal plane of the Grand Raiden spectrometer, taken at 2.5${}^{°}$. The central densely hatched region represents $\mathrm{true}+\mathrm{background}$ events. The off-center sparsely hatched regions represent only background events. The real events were obtained by subtracting background events from the $\mathrm{true}+\mathrm{background}$ events.

Figure 2

(a) Horizontal-position spectrum of the ${}^{112}\mathrm{Sn}$($\alpha$,${\alpha }^{\prime }$) reaction at 0${}^{°}$. The hatched region is background events. (b) Background-free spectrum.

Figure 3

Excitation-energy spectra obtained from inelastic $\alpha$ scattering at ${\theta }_{\mathrm{lab}}$ $=$ 0.69${}^{°}$ for all even-$A$ Sn isotopes.

Figure 4

(a) Ratio of the elastic $\alpha$-scattering cross sections to the Rutherford cross sections for ${}^{112}\mathrm{Sn}$ at 386 MeV. (b) Differential cross sections for excitation of the 2${}_1^{+}$ state in ${}^{112}\mathrm{Sn}$. The solid lines are the results of the folding-model calculations.

Figure 5

Same as Fig. 4, except for ${}^{120}\mathrm{Sn}$.

Figure 6

Same as Fig. 4, except for ${}^{124}\mathrm{Sn}$.

Figure 7

Angular distribution of 1-MeV bins centered at $E_x=16.5$ MeV for ${}^{112}\mathrm{Sn}$($\alpha$,${\alpha }^{\prime }$) and ${}^{124}\mathrm{Sn}$($\alpha$,${\alpha }^{\prime }$). The solid squares are the experimental data and the solid lines are the MDA fits to the data. Also shown are the contributions to the fits from $L=0$ (dashed line), $L=1$ (dotted line), $L=2$ (dash-dotted line), and $L=3$ (small-dashed line) multipoles, as well as from the IVGDR (dash-dot-dotted line).

Figure 8

Same as Fig. 7 except for the 25.5-MeV energy bin (see text).

Figure 9

ISGMR strength distributions obtained for the Sn isotopes in the present experiment. Error bars represent the uncertainties from fitting the angular distributions in the MDA procedure. The solid lines show Lorentzian fits to the data.

Figure 10

ISGDR strength distributions obtained for the Sn isotopes in the present experiment. Error bars represent the uncertainties from fitting the angular distributions in the MDA procedure. The solid lines show 2-Lorentzian fits to the data, with the dashed and dot-dashed lines, respectively, showing the fitted LE and HE components (see text).

Figure 11

ISGQR strength distributions obtained for the Sn isotopes in the present experiment. Error bars represent the uncertainties from fitting the angular distributions in the MDA procedure. The solid lines show Lorentzian fits to the data.

Figure 12

HEOR strength distributions obtained for the Sn isotopes in the present experiment. Error bars represent the uncertainties from fitting the angular distributions in the MDA procedure. The solid lines show 2-Lorentzian fits to the data.

Figure 13

Systematics of the moment ratios $m_1/m_0$ for the ISGMR strength distributions in the Sn isotopes. The experimental results (filled squares) are compared with results from nonrelativistic RPA calculations (without pairing) by Colò et al. [65, 66] (filled circles), relativistic calculations of Piekarewicz [67] (triangles), RMF calculations from Vretenar et al. [68] (diamonds), and QTBA calculations from the Jülich group [69] (sideways triangles). Results for ${}^{112}\mathrm{Sn}$, ${}^{116}\mathrm{Sn}$, and ${}^{124}\mathrm{Sn}$ reported by the TAMU group [19, 21] are also shown (inverted triangles).