Yesterday a paper appeared in Nature which describes on the second case of a pair of "mirror nuclei" (which differ from each other by having the number of protons and neutrons swapped) in which the ground states have different spin.
The pair in question is strontium-73 (Z=38, N=35) and bromine-73 (Z=35, N=38) which have been measured to have spin-parity assignments of 5/2– and 1/2– respectively. The states of mirror nuclei are pretty close to identical, thanks to the isospin symmetry of the nuclear force: To a good approximation the nuclear force looks the same between pairs of protons, pairs of neutrons and neutron-proton pairs. There are a couple of ways in which differences appear in mirror nuclei - e.g. because there is also the Coulomb force in play which acts between protons but not neutrons, and these small differences can sometimes cause an effect like the one seen in the Sr-Br pair. In this case, the small differences are enough to give a different ground state as there seems to be a very low-lying state close to the ground state in these nuclei and the small differences happen to be enough to swap the order of these levels in the two nuclei.
The figure to the right is part of one of the figures in the supplementary material on the paper. It's a section in the nuclear chart of isotopes in which the line of N=Z nuclei appears as a vertical line in the middle, and nuclei close to this line are shown - the ones for which mirror pairs are known to exist. The two pairs coloured in black with little cracks in, are the two cases in which the mirror ground state symmetry is broken. The other case which was previously known is 16F/16N. In that case the fluorine isotope has its last proton unbound, and it only exists as a nucleus thanks to the Coulomb barrier. The nitrogen valence proton is not unbound and this significant difference is enough to cause the difference in the ground states. The same effect is not in play in the Sr-Br case.
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