Proton-neutron pairing correlations in the self conjugate nucleus $^{38}$K probed via a direct measurement of the isomer shift

A marked difference in the nuclear charge radius was observed between the $I^{\pi}$ =3$^{+}$ ground state and the $I^{\pi}$ =0$^{+}$ isomer of $^{38}$K and is qualitatively explained using an intuitive picture of proton-neutron pairing. In a high-precision measurement of the isomer shift using bunched-beam collinear laser spectroscopy at CERN-ISOLDE, a change in the mean-square charge radius of ⟨r$^{2}_{c}$⟩( $^{38}$K$^{m}$ ) − ⟨r$^{2}_{c}$ ⟩($^{38}$K$^{g}$ )=0.100(6) fm$^2$ was obtained. This is an order of magnitude more accurate than the result of a previous indirect measurement from which it was concluded that both long-lived states in $^{38}$K have similar charge radii. Our observation leads to a substantially different understanding since the difference in charge radius is, moreover, opposite in sign to previously reported theoretical predictions. It is demonstrated that the observed isomer shift can be reproduced by large-scale shell-model calculations including proton and neutron excitations across the N, Z=20 shell gaps, confirming the significance of cross-shell correlations in the region of $^{40}$Ca .