Isoscalar $E0,E1,E2$, and $E3$ strength in ${}^{92,96,98,100}\mathrm{Mo}$

Isoscalar giant resonances in ${}^{92,96,98,100}\mathrm{Mo}$ have been studied with inelastic scattering of 240-MeV $\alpha$ particles at small angles including $0^{\circ }$. A significant fraction of the energy-weighted sum rule was found for isoscalar $E0(107\%,105\%,103\%,\mathrm{and}110\%),E1(71\%,71\%,70\%,\mathrm{and}55\%),E2(73\%,69\%,85\%,\mathrm{and}79\%)$, and high-energy octupole $E3(52\%,65\%,61\%,\mathrm{and}53\%)$ resonances in ${}^{92,96,98,100}\mathrm{Mo}$, respectively. Spherical Hartree-Fock-based random-phase approximation calculations were performed for each multipole using the KDE0v1 Skyrme-type effective interaction, and the results are compared to the experimental distributions.

Figure 1

Inelastic $\alpha$ spectra obtained at two angles for ${}^{92,96,98,100}\mathrm{Mo}$. The thick pink lines show a continuum chosen for the analysis.

Figure 2

The angular distributions of the ${}^{92}\mathrm{Mo}$ cross sections for a 500-keV wide bin centered at the excitation energy (in MeV) indicated in each panel for inelastic $\alpha$ scattering for three excitation ranges of the GR peak and the continuum. The lines through the data points indicate the multipole fits. Contributions of each multipole are shown $(L=0$ red; $L=1,T=0$ blue-green; $L=1,T=1$ blue; $L=2$ maroon; $L=3$ brown; $L=4$ green). The statistical errors are shown but in many cases are smaller than the data points.

Figure 3

The angular distributions of the ${}^{98}\mathrm{Mo}$ cross sections. See Fig. 2 caption.

Figure 4

Strength distributions obtained for ${}^{92}\mathrm{Mo}$ and ${}^{96}\mathrm{Mo}$ are shown by the histograms. Error bars represent the uncertainty due to the fitting of the angular distributions and different choices of the continuum as described in the text. The thick lines in the $E0$ and $E1$ distributions represent the individual peaks and their sum obtained from the Gaussian fits (the blue and yellow lines are the individual peaks, whereas the brown line indicates the sum). The thin (red) lines are the strength distributions obtained with the HF-RPA calculations using the KDE0v1 interaction, smeared to more closely represent the data as discussed in the text.

Figure 5

Strength distributions obtained for ${}^{98,100}\mathrm{Mo}$ are shown by the histograms. See Fig. 4 caption.

Figure 6

The centroids of the Gaussians obtained from the fits to the $E0$ distributions for the Mo isotopes are plotted versus $A$. The error bars indicate the uncertainty obtained using the errors shown in Figs. 4 and 5. The (yellow) lines show (a) $74/A^{-1/3}$ and (b) $109/A^{-1/3}$ in two plots.

Figure 7

The centroid of the Gaussian fit to the high-energy peak in the ISGDR distributions for each of the Mo isotopes is plotted versus $A$ in (a) whereas the strength of the upper peak is plotted in (b). The error bars indicate the uncertainty obtained using the errors shown in Figs. 4 and 5. The triangles (blue) show $m_1/m_0$ for the strength between $E_{\mathrm{x}}=20\mathrm{MeV}$ and $E_{\mathrm{x}}=36\mathrm{MeV}$ calculated with HF-RPA using the KDE0v1 interaction.

Figure 8

The centroid of the $E2$ strength in each of the Mo isotopes obtained in this work (red squares) is plotted versus $A$. The error bars indicate the uncertainty obtained using the errors shown in Figs. 4 and 5. Also shown with error bars are centroids reported in Ref. [1] (black diamonds) and a measurement for ${}^{92}\mathrm{Mo}$ [2] (blue circle). The light blue triangles show the centroid of the $E2$ strength obtained from HF-RPA calculations with the KDE0v1 interaction.

Figure 9

The centroid of the $E3$ strength observed above $E_{\mathrm{x}}=15\mathrm{MeV}$ for each of the Mo isotopes in this work (red squares) is plotted versus $A$ in (a), whereas the strength in this region is plotted in (b). The error bars indicate the uncertainty obtained using the errors shown in Figs. 4 and 5. The large light blue triangles show (a) the centroid and (b) the % EWSR of the $E3$ strength between $E_{\mathrm{x}}=15\mathrm{MeV}$ and $E_{\mathrm{x}}=36\mathrm{MeV}$ obtained from HF-RPA calculations with the KDE0v1 interaction.