Properties of $K^{\pi }=0_1^{+}$, $K^{\pi }=2^{-}$, and $K^{\pi }=0_1^{-}$ bands of ${}^{20}\mathrm{Ne}$ probed via proton and $\alpha$ inelastic scattering

The $K^{\pi }=0_1^{+}$, $K^{\pi }=2^{-}$, and $K^{\pi }=0_1^{-}$ bands of ${}^{20}\mathrm{Ne}$ are investigated with microscopic structure and reaction calculations via proton and $\alpha$ inelastic scattering off ${}^{20}\mathrm{Ne}$. Structures of ${}^{20}\mathrm{Ne}$ are calculated with variation after parity and total angular momentum projections in the antisymmetrized molecular dynamics (AMD). The $K^{\pi }=0_1^{+}$ and $K^{\pi }=0_1^{-}$ bands have ${}^{16}\mathrm{O}+\alpha$ cluster structures, whereas the $K^{\pi }=2^{-}$ band shows a ${}^{12}\mathrm{C}+2\alpha$-like structure. Microscopic coupled-channel calculations of proton and $\alpha$ scattering off ${}^{20}\mathrm{Ne}$ are performed by using the proton-nucleus and $\alpha$-nucleus potentials, which are derived by folding the Melbourne $g$-matrix $NN$ interaction with AMD densities of ${}^{20}\mathrm{Ne}$. The calculation reasonably reproduces the observed cross sections of proton scattering at $E_p=25$–35 MeV and $\alpha$ scattering at $E_{\alpha }=104$–386 MeV. Transition properties from the ground to excited states are discussed by reaction analyses of proton and $\alpha$ inelastic processes. Mixing of the $K^{\pi }=2^{-}$ and $K^{\pi }=0_1^{-}$ bands is discussed by detailed analysis of the $0_1^{+}\to 3_1^{-}$ and $0_1^{+}\to 3_2^{-}$ transitions. For the $3_1^{-}$ state, mixing of the $K^{\pi }=0_1^{-}$ cluster component in the $K^{\pi }=2^{-}$ band plays an important role in the transition properties.

Figure 1

The calculated energy levels of ${}^{20}\mathrm{Ne}$ obtained by (a) AMD, (b) AMD-ls34, and (c) CM, and (d) the experimental levels assigned to the $K^{\pi }=0_1^{+}$, $K^{\pi }=2^{-}$, $K^{\pi }=0_1^{-}$, and $K^{\pi }=0_{\mathrm{hn}}^{+}$ bands.

Figure 2

Density distribution of intrinsic wave functions for the bandhead states (a) $0_1^{+}$, (b) $2_1^{-}$, and (c) $1_1^{-}$ of the $K^{\pi }=0_1^{+}$, $K^{\pi }=2^{-}$, and $K^{\pi }=0_1^{-}$ bands of ${}^{20}\mathrm{Ne}$ obtained with AMD. The densities projected onto $X-Z$, $Y-Z$, and $Y-X$ planes are shown in left, middle, and right panels, respectively. Intrinsic axes are chosen as $\langle ZZ\rangle \ge \langle YY\rangle \ge \langle XX\rangle$ and $\langle XY\rangle =\langle YZ\rangle =\langle ZX\rangle =0$.

Figure 3

Matter densities of ${}^{20}\mathrm{Ne}$ calculated with AMD.

Figure 4

Elastic and inelastic form factors of ${}^{20}\mathrm{Ne}$ calculated with AMD (red solid lines) and CM (green dashed lines) compared with the experimental data (circles). The $3_1^{-}$ and $3_2^{-}$ form factors calculated with AMD-ls34 (blue dotted lines) are also shown in panels (e) and (f), respectively. The theoretical $2_1^{+}$ form factors obtained by AMD (CM) are renormalized by $f^{\mathrm{tr}}=0.97$ (1.11). The theoretical $3_1^{-}$ form factors obtained by AMD and AMD-ls34 are multiplied by $f^{\mathrm{tr}}=2.22$ and 1.30, respectively. The experimental data are those measured by electron scattering [57].

Figure 5

Transition densities of ${}^{20}\mathrm{Ne}$ calculated with AMD and CM. Theoretical values obtained by AMD (CM) for the $2_1^{+}$ state are renormalized by $f^{\mathrm{tr}}=0.97$ (1.11), and those for the $3_1^{-}$ state are multiplied by $f^{\mathrm{tr}}=2.22$.

Figure 6

Cross sections of proton elastic and inelastic scattering off ${}^{20}\mathrm{Ne}$ at incident energies of $E_p=25$, 30, and 35 MeV calculated with $\mathrm{MCC}+\mathrm{AMD}$ (red solid lines), $\mathrm{DWBA}+\mathrm{AMD}$ (blue dotted lines), and $\mathrm{MCC}+\mathrm{CM}$ (green dashed lines), which are labeled as AMD, DWBA, and CM, respectively. Experiment data are cross sections at $E_p=24.5$ MeV [26, 28], 30 MeV [28], and 35 MeV [58].

Figure 7

Cross sections of $\alpha$ elastic and inelastic scattering off ${}^{20}\mathrm{Ne}$ at incident energies of $E_{\alpha }=104$, 146, and 386 MeV calculated with $\mathrm{MCC}+\mathrm{AMD}$ (red solid lines), $\mathrm{DWBA}+\mathrm{AMD}$ (blue dotted lines), and $\mathrm{MCC}+\mathrm{CM}$ (green dashed lines), which are labeled as AMD, DWBA, and CM, respectively. Experiment data are cross sections at $E_{\alpha }=$ 104 MeV [30], and 146 MeV [59], and 386 MeV [31]. The $E_{\alpha }=386$ MeV data from Ref. [31] are multiplied by a factor of 2. The $(\alpha ,{\alpha }^{\prime })$ data at $E_{\alpha }=104$ MeV in the panels (d) and (e) are the cross sections observed for $E_x=5.7$ MeV and may contain $1_1^{-}$(5.79 MeV) and $3_1^{-}$(5.62 MeV) contributions.

Figure 8

Transition densities from the ground to $3^{-}$ states obtained with AMD (default) and AMD-ls34 (modified spin-orbit strength): (a) transition densities to the $3_1^{-}$ state, (b) those but $r^2$ weighted, (b) transition densities to the $3_2^{-}$ state. Transition densities to the $3_1^{-}$ state in (a) and (b) are are renormalized by $f_{\mathrm{tr}}=2.22$ for AMD and 1.30 for AMD-ls34.

Figure 9

Cross sections of proton and $\alpha$ inelastic scattering calculated with MCC using the AMD (default) and AMD-ls34 (modified spin-orbit strength) densities, and the experimental cross sections. The $(p,p^{\prime })$ cross sections at $E_p=25$ and 35 MeV to the (a) $3_1^{-}$ and (b) $3_2^{-}$ states and the $(\alpha ,{\alpha }^{\prime })$ cross sections at $E_{\alpha }=146$ and 386 MeV to the (c) $3_1^{-}$ and (d) $3_2^{-}$ states. The $(p,p^{\prime })$ data at $E_p=24.5$ MeV [28] and $(\alpha ,{\alpha }^{\prime })$ data at $E_{\alpha }=104$ MeV [30] and 386 MeV [31] are also shown. The $(\alpha ,{\alpha }^{\prime })$ data at 386 MeV of Ref. [31] are multiplied by a factor of 2. The $(\alpha ,{\alpha }^{\prime })$ data at $E_{\alpha }=104$ MeV in the panel (c) are not cross sections for an individual state but may contain $1_1^{-}$ and $3_1^{-}$ contributions around 5.7 MeV.