Giant resonances in ${}^{46,48}\mathrm{Ti}$

The giant resonance region from 9 MeV $<E_x<55$ MeV in ${}^{46}\mathrm{Ti}$ and ${}^{48}\mathrm{Ti}$ has been studied with inelastic scattering of 240 MeV $\alpha$ particles at small angles including $0^{°}$. Isoscalar monopole strength in ${}^{46}\mathrm{Ti}$ (${}^{48}\mathrm{Ti}$) was found corresponding to have $71+15/-12\%$ ($96+14/-12\%$) of the $E0$ energy weighted sum rule (EWSR) with a centroid of $18.66+0.65/-0.25\%$ MeV ($18.80+0.45/-0.18\%$ MeV), respectively. In ${}^{46}\mathrm{Ti}$ (${}^{48}\mathrm{Ti}$), $46\pm 12\%$ ($56\pm 12\%$) of the $E1$, and $60\pm 11\%$ ($87\pm 11\%$) of the $E2$ EWSR were identified.

Figure 1

The angular distributions of the differential cross sections for $\alpha$ particle elastic scattering and scattering to the first excited states for ${}^{46}\mathrm{Ti}$ and ${}^{48}\mathrm{Ti}$. The inelastic scattering cross sections are shown multiplied by 0.1. The lines in both plots are the fits for the elastic scattering and calculated inelastic scattering cross sections using the reduced electric quadrupole transition probability $B(E$2) for the first excited states.

Figure 2

Excitation energy spectra for ${}^{46,48}\mathrm{Ti}$. Experimental data are shown by the histograms. The solid lines in the figures show the range of continuum curves chosen for the analysis.

Figure 3

Angular distributions of the cross sections for the GR peak and the continuum in ${}^{46}\mathrm{Ti}$. The excitation energies are the center energies of excitation energy bins which are 0.64 MeV wide. Experimental data are shown by solid squares. Each experimental point has an error bar from the statistics and standard deviation where more than one data point was available. The results of the multipole fits are shown along with each multipole, $L=0$, 1, 2, 3, and IVGDR components shown by the thick solid, dashed, dot, dashed-dot, and solid lines, respectively.

Figure 4

Angular distributions of the cross sections for the GR peak and the continuum in ${}^{48}\mathrm{Ti}$. The excitation energies are the center energies of excitation energy bins which are 0.64 MeV wide. Experimental data are shown by solid squares. Each experimental point has an error bar from the statistics and standard deviation where more than one data point was available. The results of the multipole fits are shown along with each multipole, $L=0$, 1, 2, 3, 4 and IVGDR components shown by the thick solid, dashed, dot, dashed-dot, dashed-double-dot, and solid lines, respectively.

Figure 5

Strength distributions obtained for ${}^{46,48}\mathrm{Ti}$ 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 smooth lines show Gaussian fits.

Figure 6

GMR energies $(E_{\mathrm{GMR}}=(m_3/m_1){}^{1/2})$ as a function of mass number for nuclei with $A=40\text{−}58$. The solid circles represent the experimental data. The open circles show results obtained using the extended Thomas-Fermi (ETF) approximation with a Skyrme interaction (SkM${}^{*}$) [7] and solid squares show results obtained using the relativistic mean-field (RMF) theory with the NLC interaction [32]. The lines are added to guide the eye.