$\Delta l =1$ coupling of single-particle orbitals in octupole deformed nuclei

Conventionally, octupole deformation in nuclei has been attributed to strong $\Delta l=3$ couplings between opposite-parity single-particle orbitals. In this work, we demonstrate that the often-overlooked $\Delta l=1$ mode also plays an important role. Taking orbitals near the octupole magic number $N = 134$ as a benchmark, we systematically evaluate the $\Delta l = 1$ and $\Delta l = 3$ mixing ratios of the wave functions within the Nilsson model, interpreting the trends through matrix elements of the deformed potential. We introduce component-resolved single-particle octupole energy contributions, based on the Hellmann--Feynman relation, to quantify the contributions of each $(\Delta l,\Delta j)$ coupling. Furthermore, the impact of $\Delta l = 1$ coupling on the rotational structure is demonstrated via particle-rotor model calculations for $^{221}$Ra and $^{223}$Th. Our work suggests that $\Delta l=1$ and $\Delta l=3$ octupole couplings act synergistically in driving reflection asymmetry, necessitating a revised paradigm for understanding octupole correlation.