Symmetry properties of pair correlations in heavy deformed nuclei

Nucleons are known to form pairing correlations with various types of spin-symmetries. Spin-singlet neutron-neutron and proton-proton pairing is abundant in the nuclear chart but spin-triplet and mixed-spin proton-neutron pairing correlations have also been predicted to form at least in the ground states of certain nuclei. A realistic candidate region is that of the lightest Lanthanides where it was recently demonstrated that the nuclear deformation expected to emerge enhances spin-triplet pairing correlations. In this paper we provide the details of the deformed multimodal Hartree-Fock-Bogolyubov theory that lead to this conclusion, as well as the details of the effects identified. We present in detail the response of different pairing correlations to various deformation modes and calculate their signatures in the odd-even staggering of masses. This paper provides a detailed discussion, and some resolutions, on the long-standing question ``what is the effect of nuclear deformation on the various pairing correlations?''