Transition properties of low-lying states in ${}^{28}\mathrm{Si}$ probed via inelastic proton and $\alpha$ scattering

$0^{+}, 1^{-}, 2^{+}$, and $3^{-}$ excitations of ${}^{28}\mathrm{Si}$ are investigated via proton and $\alpha$ inelastic scattering off ${}^{28}\mathrm{Si}$. The structure calculation of ${}^{28}\mathrm{Si}$ is performed with the energy variation after total angular momentum and parity projections in the framework of antisymmetrized molecular dynamics (AMD). As a result of the AMD calculation, the oblate ground and prolate bands, $0^{+}$ and $3^{-}$ excitations, and the $1^{-}$ and $3^{-}$ states of the $K^{\pi }=0^{-}$ band are obtained. Using the matter and transition densities of ${}^{28}\mathrm{Si}$ obtained by AMD, microscopic coupled-channels calculations of proton and $\alpha$ scattering off ${}^{28}\mathrm{Si}$ are performed. The proton-${}^{28}\mathrm{Si}$ potentials in the reaction calculation are microscopically derived by folding the Melbourne $g$-matrix $NN$ interaction with the AMD densities of ${}^{28}\mathrm{Si}$. The $\alpha -{}^{28}\mathrm{Si}$ potentials are obtained by folding the nucleon-${}^{28}\mathrm{Si}$ potentials with an $\alpha$ density. The calculation reasonably reproduces the observed elastic and inelastic cross sections of proton and $\alpha$ scattering. Transition properties are discussed by combining the reaction analysis of proton and $\alpha$ scattering and structure features such as transition strengths and form factors. The isoscalar monopole and dipole transitions are focused.

Figure 1

Density distribution of intrinsic wave functions before the parity and angular-momentum projections for the $0_1^{+}, 0_{\mathrm{vib}}^{+}, 0_{\mathrm{pro}}^{+}, 1_{\mathrm{IS}1}^{-}$, and $3_1^{-}$ states of ${}^{28}\mathrm{Si}$ obtained by AMD. The density projected onto $X-Z, Y-Z$, and $Y-X$ planes are shown in the left, middle, and right panels, respectively. Here, intrinsic axes are chosen as $\langle ZZ\rangle \ge \langle YY\rangle \ge \langle XX\rangle$ and $\langle XY\rangle =\langle YZ\rangle =\langle ZX\rangle =0$. The deformation parameters $(\beta ,\gamma )$ calculated from the values of $\langle ZZ\rangle , \langle YY\rangle$, and $\langle XX\rangle$ are shown in each panel.

Figure 2

Energy spectra of ${}^{28}\mathrm{Si}$. (left) Calculated energy spectra of the ground and prolate bands, $0_{\mathrm{vib}}^{+}$ and $3_1^{-}$ excitations of the ground band, and the $K^{\pi }=0^{-}$ band of the $1_{\mathrm{IS}1}^{-}$ and $3_2^{-}$ states. (right) Experimental spectra corresponding to the theoretical states.

Figure 3

Matter densities of the $0^{+}, 2^{+}, 1^{-}$, and $3^{-}$ states of ${}^{28}\mathrm{Si}$ calculated with AMD.

Figure 4

Elastic and inelastic form factors of positive-parity states of ${}^{28}\mathrm{Si}$. The inelastic form factors obtained by AMD are renormalized by $f_{\mathrm{tr}}^2$ with the factor $f_{\mathrm{tr}}$ listed in Table 2. The experimental data are those measured by electron scattering in Refs. [9, 10, 14].

Figure 5

Inelastic form factors of negative-parity states of ${}^{28}\mathrm{Si}$. The inelastic form factors obtained by AMD are renormalized by $f_{\mathrm{tr}}^2$ with the factor ($f_{\mathrm{tr}}$) listed in Table 2. The experimental data measured by electron scattering are from Refs. [9, 10, 14]. In panel (a) for the $3_1^{-}$ state, triangles indicate the experimental data of the sum of the $3_1^{-}$ (6.879 MeV) and $4_2^{+}$ (6.888 MeV) contributions, and circles indicate the $3_1^{-}$ (6.879 MeV) data evaluated by subtracting the $4_2^{+}$ (6.888 MeV) contribution from the sum [9].

Figure 6

Transition densities of rank $\lambda =J$ transitions from the ground state to $J^{\pi }$ states. The theoretical values calculated with AMD are renormalized by $f_{\mathrm{tr}}$ listed in Table 2.

Figure 7

Cross sections of elastic proton and $\alpha$ scattering off ${}^{28}\mathrm{Si}$ calculated with the CC calculation for proton incident energies $E_p=65$, 100, 180 MeV and $\alpha$ incident energies $E_{\alpha }=120$, 130, 240, and 386 MeV. The experimental data are $(p,p)$ cross sections at $E_p=65$ MeV [13], 100 MeV [58], and 180 MeV [14], and $(\alpha ,\alpha )$ cross sections at $E_{\alpha }=120$ MeV [59, 60], 240 MeV [19], and 386 MeV [29].

Figure 8

Proton inelastic-scattering cross sections at incident energies $E_p=65$, 100, 180 MeV obtained by the CC and DWBA calculations. Experimental data are cross sections at $E_p=65$ MeV [13], $E_p=100$ MeV [12], $E_p=180$ MeV [14], and $E_p=185$ MeV [11]. For the experimental data of the $0_3^{+}$ cross sections, the upper limit of the $0_3^{+}$ cross section at ${\theta }_{\mathrm{c}.\mathrm{m}.}=4^{\circ }$ from Ref. [11] and a quarter of the $0_2^{+}$ cross section. at ${\theta }_{\mathrm{c}.\mathrm{m}.}={20}^{\circ }$ evaluated from the observed spectrum shown in Ref. [14] are shown. See text.

Figure 9

$\alpha$ inelastic-scattering cross sections at incident energies $E_{\alpha }=120$, 130, 240, and 400 MeV obtained by the CC and DWBA calculations. Experiment data are $(\alpha ,{\alpha }^{\prime })$ cross sections at $E_{\alpha }=120$ MeV [59], 130 MeV [32], 240 MeV [19], and 386 MeV [32].