Evolution of the E(1/2(1)(+))-E(3/2(1)(+)) energy spacing in odd-mass K, Cl, and P isotopes for N=20-28

The energy of the first excited state in the neutron-rich $N=28$ nucleus $^{45}\mathrm{Cl}$ has been established via in-beam \ensuremath{\gamma}-ray spectroscopy following proton removal. This energy value completes the systematics of the $E(1/{2}_{1}^{+})\ensuremath{-}E(3/{2}_{1}^{+})$ level spacing in odd-mass K, Cl, and P isotopes for $N=20\text{\ensuremath{-}}28$. The results are discussed in the framework of shell-model calculations in the $\mathit{sd}\text{\ensuremath{-}}\mathit{fp}$ model space. The contribution of the central, spin-orbit, and tensor components is discussed from a calculation based on a proton single-hole spectrum from $G$-matrix and $\ensuremath{\pi}+\ensuremath{\rho}$ meson exchange potentials. A composite model for the proton $0{d}_{3/2}\ensuremath{-}1{s}_{1/2}$ single-particle energy shift is presented.