We report on the first γ-ray spectroscopy of 51,53K produced via the 52,54Ca(p,2p) reactions at ∼250 MeV/nucleon. Unambiguous final-state angular-momentum assignments were achieved for beam intensities down to few particles per second by using a new technique based on reaction vertex tracking combined with a thick liquid-hydrogen target. Through γ-ray spectroscopy and exclusive parallel momentum distribution analysis, 3/2+ ground states and 1/2+ first excited states in 51,53K were established quantifying the natural ordering of the 1 d 3 / 2 and 2 s 1 / 2 proton-hole states that are restored at N = 32 and 34. State-of-the-art ab initio calculations and shell-model calculations with improved phenomenological effective interactions reproduce the present data and predict consistently the increase of the E(1/2 1 + ) - E(3/2 1 + ) energy differences towards N = 40.
