The triaxial and hexadecapole deformations of the $K^\pi=0^+$ and $K^\pi=2^+$ bands of $^{24}$Mg have been investigated by the inelastic scatterings of various probes, including electrons, protons, and alpha($\alpha$) particles, for a prolonged time. However, it has been challenging to explain the unique properties of the scatterings observed for the $4^+_1$ state through reaction calculations. This paper investigates the structure and transition properties of the $K^\pi=0^+$ and $K^\pi=2^+$ bands of $^{24}$Mg employing the microscopic structure and reaction calculations via inelastic proton and $\alpha$ scattering. In particular, the $E4$ transitions to the $4^+_1$ and $4^+_2$ states are reexamined. The structure of $^{24}$Mg was calculated employing the variation after the parity and total angular momentum projections in the framework of the antisymmetrized molecular dynamics (AMD). The inelastic proton and $\alpha$ reactions were calculated by the microscopic coupled-channel (MCC) approach by folding the Melbourne $g$-matrix $NN$ interaction with the AMD densities of $^{24}$Mg. Reasonable results were obtained on the properties of the structure, including the energy spectra and $E2$ and $E4$ transitions of the $K^\pi=0^+$ and $K^\pi=2^+$ bands owing to the enhanced collectivity of triaxial deformation. The MCC+AMD calculation successfully reproduced the angular distributions of the $4^+_1$ and $4^+_2$ cross sections of proton scattering at incident energies of $E_p=40$–100 MeV and $\alpha$ scattering at $E_\alpha=100$–400 MeV. This is the first microscopic calculation to describe the unique properties of the $0^+_1\to 4^+_1$ transition. In the inelastic scattering to the $4^+_1$ state, the dominant two-step process of the $0^+_1\to 2^+_1\to 4^+_1$ transitions and the deconstructive interference in the weak one-step process were essential.

中文翻译：

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质子和 α 非弹性散射到 $^{24}\textrm{Mg}$ 的 $4^+_1$ 和 $4^+_2$ 状态的微观耦合通道计算

$^{24}$Mg 的 $K^\pi=0^+$ 和 $K^\pi=2^+$ 带的三轴和十六极变形已经通过各种探针的非弹性散射进行了研究，包括电子、质子和 alpha($\alpha$) 粒子，持续很长时间。然而，通过反应计算来解释 $4^+_1$ 状态下观察到的散射的独特性质一直具有挑战性。本文利用非弹性质子的微观结构和反应计算研究了$^{24}$Mg的$K^\pi=0^+$和$K^\pi=2^+$能带的结构和跃迁性质。和 $\alpha$ 散射。特别是，重新检查 $E4$ 到 $4^+_1$ 和 $4^+_2$ 状态的转换。$^{24}$Mg 的结构是在反对称分子动力学（AMD）的框架内利用奇偶校验和总角动量投影后的变化计算的。非弹性质子和 $\alpha$ 反应通过微观耦合通道 (MCC) 方法通过折叠墨尔本 $g$-矩阵 $NN$ 相互作用与 $^{24}$Mg 的 AMD 密度来计算。在结构性质上得到了合理的结果，包括能谱和$K^\pi=0^+$和$K^\pi=2^+$能带的$E2$和$E4$跃迁，由于三轴变形的整体性增强。MCC+AMD 计算成功地再现了在 $E_p=40$–100 MeV 入射能量和 $E_\阿尔法=100$–400 MeV。这是第一个描述 $0^+_1\to 4^+_1$ 转变独特性质的微观计算。在非弹性散射到$4^+_1$状态下，$0^+_1\to 2^+_1\to 4^+_1$跃迁占主导地位的两步过程和弱一步过程中的解构干涉是必不可少的。