Neutron dominance in excited states of ${}^{26}\mathrm{Mg}$ and ${}^{10}\mathrm{Be}$ probed by proton and $\alpha$ inelastic scattering

Isospin characters of nuclear excitations in ${}^{26}\mathrm{Mg}$ and ${}^{10}\mathrm{Be}$ are investigated via proton ($p$) and alpha ($\alpha$) inelastic scattering. A structure model of antisymmetrized molecular dynamics (AMD) is applied to calculate the ground and excited states of ${}^{26}\mathrm{Mg}$ and ${}^{10}\mathrm{Be}$. The calculation describes the isoscalar feature of the ground-band $2_1^{+}(K^{\pi }=0_1^{+}$) excitation and predicts the neutron dominance of the side-band $2_2^{+}(K^{\pi }=2^{+}$) excitation in ${}^{26}\mathrm{Mg}$ and ${}^{10}\mathrm{Be}$. The $p$ and $\alpha$ inelastic scattering off ${}^{26}\mathrm{Mg}$ and ${}^{10}\mathrm{Be}$ is calculated by microscopic coupled-channel (MCC) calculations with a $g$-matrix folding approach by using the matter and transition densities of the target nuclei calculated with AMD. The calculation reasonably reproduces the observed $0_1^{+},2_1^{+}$, and $2_2^{+}$ cross sections of ${}^{26}\mathrm{Mg}+p$ scattering at incident energies $E_p=24$ and 40 MeV and of ${}^{26}\mathrm{Mg}+\alpha$ scattering at $E_{\alpha }=104$ and 120 MeV. For ${}^{10}\mathrm{Be}+p$ and ${}^{10}\mathrm{Be}+\alpha$ scattering, inelastic cross sections to the excited states in the $K^{\pi }=0_1^{+}$ ground, $K^{\pi }=2^{+}$ side, $K^{\pi }=0_2^{+}$ cluster, and $K^{\pi }=1^{-}$ cluster bands are investigated. The isospin characters of excitations are investigated via inelastic scattering processes by comparison of the production rates in the ${}^{10}\mathrm{Be}+p,{}^{10}\mathrm{Be}+\alpha$, and ${}^{10}\mathrm{C}+p$ reactions. The result predicts that the $2_2^{+}$ state is selectively produced by the ${}^{10}\mathrm{Be}+p$ reaction because of the neutron dominance in the $2_2^{+}$ excitation as in the case of the ${}^{26}\mathrm{Mg}+p$ scattering to the $2_2^{+}$ state, whereas its production is significantly suppressed in the ${}^{10}\mathrm{C}+p$ reaction.

Figure 1

Energy levels of ${}^{26}\mathrm{Mg}$ and ${}^{10}\mathrm{Be}$. (a) Calculated ${}^{26}\mathrm{Mg}$ levels of the $K^{\pi }=0_1^{+}$ ground and $K^{\pi }=2^{+}$ side bands compared with the experimental levels. In the experimental spectra, candidate states for $3^{+}$ and $4^{+}$ states of band members are shown. (b) Calculated ${}^{10}\mathrm{Be}$ levels [46] of the $K^{\pi }=0_1^{+}$ ground, $K^{\pi }=2^{+}$ side, $K^{\pi }=0_2^{+}$ cluster, and $K^{\pi }=1^{-}$ cluster bands are shown together with the observed energy levels. The experimental data are from Refs. [52, 53].

Figure 2

Charge form factors of ${}^{26}\mathrm{Mg}$. The inelastic form factors $F\left(q\right)$ obtained by AMD are renormalized by $f_p^{\text{tr}}$ given in Table 4. The results of the $0_1^{+},2_1^{+},2_2^{+},4_{\text{gs}}^{+}$, and $4_{K2}^{+}$ states are compared with the experimental data for the $0_1^{+}$ state from Ref. [56] and those for the $2_1^{+}$(1.81 MeV), $2_2^{+}$(2.94 MeV), $4_2^{+}$(4.90 MeV), and $4_4^{+}$(5.72 MeV) states from Ref. [55].

Figure 3

Proton and neutron diagonal and transition densities of ${}^{26}\mathrm{Mg}$. (a) The diagonal densities of the $0_1^{+}$ state. (b) The renormalized transition densities from the $0_1^{+}$ state to the $2_1^{+}$ and $2_2^{+}$ states. (c) The renormalized transition densities from the $0_1^{+}$ state to the $4_{\text{gs}}^{+}$ and $4_{K2}^{+}$ states.

Figure 4

Radial dependences and volume integrals of the real $\left[V\right(r\left)\right]$ and imaginary $\left[W\right(r\left)\right]$ parts of the present microscopic potentials (pres.) for $p+{}^{26}\mathrm{Mg}$ and $\alpha +{}^{26}\mathrm{Mg}$ elastic scattering compared with phenomenological (phen.) potentials of global optical potentials for $p$-nucleus [57, 58] and $\alpha$-nucleus [59] elastic scattering. (a), (b) $p+{}^{26}\mathrm{Mg}$ potentials at $E_p=40$ MeV and (d), (e) $\alpha +{}^{26}\mathrm{Mg}$ potentials at $E_{\alpha }=104$ MeV. In each panel, rms radii ($R_V$ and $R_W$) of the potentials are shown in the unit of fm. (c), (f) Energy dependences of the volume integrals per interacting nucleon pair defined as $J_V\equiv -\int V\left(r\right)d\mathit{r}/\left(A_p{A}_t\right)$ and $J_W\equiv -\int W\left(r\right)d\mathit{r}/\left(A_p{A}_t\right)$ ($A_p$ and $A_t$ are the mass numbers of projectile and target nuclei).

Figure 5

Cross sections of $p$ elastic and inelastic scattering off ${}^{26}\mathrm{Mg}$ at $E_p=24$, 40, 60, and 100 MeV. The results obtained by the MCC and DWBA calculations are shown by red solid and blue dashed lines, respectively. Experimental data are cross sections at $E_p=24$ MeV [12] and 40 MeV [13] from the EXFOR database [60]. (a), (b), and (c) show the calculated and experimental cross sections of the $0_1^{+},2_1^{+}$, and $2_2^{+}$ states, respectively. (d) shows the calculated $4_{\text{gs}}^{+}$ cross sections together with the data observed for the $4_2^{+}$(4.90 MeV) and $4_1^{+}$(4.32 MeV) states. (e) shows the calculated $4_{K2}^{+}$ cross sections compared with the data observed for the $4_4^{+}$(5.47 MeV) and $4_3^{+}$(5.72 MeV) states.

Figure 6

Same as Fig. 5 but for $\alpha$ scattering at $E_{\alpha }=104$, 120, 240, and 400 MeV. Experimental data at $E_{\alpha }=104$ MeV [62] from the EXFOR database [60] and $E_{\alpha }=120$ MeV [14] are shown.

Figure 7

$2_2^{+}$ cross sections of the ${}^{26}\mathrm{Mg}+p$ and ${}^{26}\mathrm{Mg}+\alpha$ reactions calculated by MCC using the renormalized transition densities with the default, case 1, and case 2 scaling. (a) $(p,p^{\prime })$ cross sections at $E_p=24$ and 40 MeV and (b) $(\alpha ,{\alpha }^{\prime })$ cross sections at $E_{\alpha }=104$ and 120 MeV. The experimental data of $(p,p^{\prime })$ are from Refs. [12, 13, 60], those of $(\alpha ,{\alpha }^{\prime })$ are from Refs. [14, 60, 62].

Figure 8

Diagonal densities of ${}^{10}\mathrm{Be}$. The proton and neutron densities of the (a) $0_{1,2}^{+}$ and (b) $1_1^{-}$ states.

Figure 9

Transition densities of ${}^{10}\mathrm{Be}$. The proton and neutron transition densities from the $0_1^{+}$ state to the (a) $2_{1,2}^{+}$, (b) $0_2^{+},2_3^{+}$, (c) $1_1^{-}$, and (d) $3_1^{-}$ states. The $0_1^{+}\to 2_1^{+}$ transition densities are renormalized ones.

Figure 10

Radial dependences and volume integrals of the real $\left[V\right(r\left)\right]$ and imaginary $\left[W\right(r\left)\right]$ parts of the present (pres.) microscopic potential for $p+{}^{10}\mathrm{Be}$ elastic scattering compared with a phenomenological potential of global optical potential [57, 58]. (a), (b) $p+{}^{10}\mathrm{Be}$ potentials at $E=60$ MeV/u. (c) Energy dependence of volume integrals $J_{V,W}$ of the potentials. In (a) and (b), rms radii ($R_V$ and $R_W$) of the potentials are shown in the unit of fm.

Figure 11

(a) Elastic and (b)–(g) inelastic cross sections of the ${}^{10}\mathrm{Be}+p$ and ${}^{10}\mathrm{C}+p$ reactions for at $E=25$, 45, 60, and 100 MeV/u. The elastic cross sections of the ${}^{10}\mathrm{Be}+p$ reaction at $E=40$ MeV/u are also shown. The MCC and DWBA results of ${}^{10}\mathrm{Be}+p$ are shown by red solid and blue dashed lines, respectively. The MCC results of ${}^{10}\mathrm{C}+p$ are shown by green long-dashed lines. The experimental ${}^{10}\mathrm{Be}+p$ elastic cross sections at $E=39.1$ MeV/u [64] and $E=59.4$ MeV/u [63] are shown by red squares in (a). The experimental ${}^{10}\mathrm{C}+p$ cross sections at $E=45$ MeV/u [60, 65] observed for the $0_1^{+}$ and $2_1^{+}$ states are shown by blue circles in (a) and (b).

Figure 12

Calculated (a) elastic and (b)–(g) inelastic cross sections of ${}^{10}\mathrm{Be}+\alpha$ at $E=25$, 68, and 100 MeV/u. The MCC and DWBA cross sections are shown by red solid and blue dashed lines, respectively. In (a) and (b), the calculated cross sections of ${}^{10}\mathrm{C}+\alpha$ are shown by green dashed lines compared with the experimental cross sections at $E=68$ MeV/u from Ref. [28].

Figure 13

Integrated cross sections to (a) the $2_1^{+}$ state and (b)–(f) other excited states of ${}^{10}\mathrm{Be}+p,{}^{10}\mathrm{C}+p$, and ${}^{10}\mathrm{Be}+\alpha$ inelastic processes are shown by (blue) triangles, (magenta) squares, and (red) circles, respectively. The cross sections at $E=25$, 60, and 100 MeV/u are calculated with MCC. In panels for the (b) $2_2^{+}$, (c) $2_3^{+}$, (d) $0_2^{+}$, (e) $1_1^{-}$, and (f) $3_1^{-}$ states, 7%–10% of the $2_1^{+}$ cross sections of the ${}^{10}\mathrm{Be}+p$ and ${}^{10}\mathrm{Be}+\alpha$ reactions are shown by light-green and pink shaded areas, respectively, as references.

Figure 14

The $0_2^{+},1_1^{-}$, and $2_2^{+}$ cross sections of the ${}^{10}\mathrm{Be}+p,{}^{10}\mathrm{C}+p$, and ${}^{10}\mathrm{Be}+\alpha$ reactions obtained by the MCC calculations. (a)${}^{10}\mathrm{Be}+p$ at $E=60$ MeV/u, (b)${}^{10}\mathrm{C}+p$ at $E=60$ MeV/u, and (c)${}^{10}\mathrm{Be}+\alpha$ at $E=68\mathrm{MeV}/\mathrm{u}$. The $0_2^{+},1_1^{-}$, and $2_2^{+}$ cross sections and their incoherent sum are shown by magenta dot-dashed, blue dashed, green long-dashed, and red solid lines.