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Description
The nuclear shell model is extremely effective in describing the behaviour of stable magic nuclei. Modern radioactive ion beam facilities have facilitated tests of the shell model along chains of magic isotopes stretching far from the valley of stability. With increased isospin asymmetry, the energies of proton and neutron orbitals can shift, in some cases moving far enough that magic numbers vanish or appear. Double-magicity of calcium isotopes has been established at conventionally non-magic $N=32,34$ on the basis of mass, radius, $E(2^+)$ and $B(E2)$. In the neighbouring elements, however, clear evidence for the persistence of these magic neutron numbers remains elusive, and more detailed spectroscopy is needed to understand how the neutron orbitals depend on the occupancy of the proton orbitals.
The $N=32,34$ shell gaps in $^{52}$Ca are thought to appear due to the 𝜈f$_{5/2}$ orbital sitting well above the 𝜈p$_{1/2}$ orbital to leave gaps on either side. One explanation for the disappearance of the $N=34$ magic number at larger $Z$ is that the when the 𝜋f$_{7/2}$ orbital starts filling, the tensor force coupling between the 𝜋f$_{7/2}$ protons and 𝜈f$_{5/2}$ neutrons reduces the energy of the 𝜈f$_{5/2}$ orbital. To experimentally probe this phenomenon we study titanium isotopes, two protons above calcium. We present results from in-beam gamma-ray spectroscopy of one-neutron knockout from $^{56,58}$Ti with the HiCARI gamma array and the BigRIPS and ZeroDegree spectrometers. Using the parallel momentum distribution of the outgoing ions, we can determine the angular momentum of the orbital from which the neutron is removed. Combined with the spectroscopic analysis, we aim to determine the relative position of the 𝜈f$_{5/2}$ with two protons in the 𝜋f$_{7/2}$ orbital, and identify whether it has already dropped below the 𝜈p$_{1/2}$ orbital.