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We utilize a nuclear shell model Hamiltonian with only two adjustableparameters to generate, for the first time, exact solutions for pairingcorrelations for light to medium-mass nuclei, including the challengingproton-neutron pairs, while also identifying the primary physics involved. Inaddition to single-particle energy and Coulomb potential terms, the shell modelHamiltonian consists of an isovector $T=1$ pairing interaction and an averageproton-neutron isoscalar $T=0$ interaction, where the $T=0$ term describes theaverage interaction between non-paired protons and neutrons. This Hamiltonianis exactly solvable, where, utilizing 3 to 7 single-particle energy levels, wereproduce experimental data for 0$^+$ state energies for isotopes with mass$A=10$ through $A=62$ exceptionally well including isotopes from He to Ge.Additionally, we isolate effects due to like-particle and proton-neutronpairing, provide estimates for the total and proton-neutron pairing gaps, andreproduce $N$ (neutron) = $Z$ (proton) irregularity. These results provide afurther understanding for the key role of proton-neutron pairing correlationsin nuclei, which is especially important for waiting-point nuclei on therp-path of nucleosynthesis.

Exact isovector pairing in a shell-model framework: Role of proton-neutron correlations in isobaric analog states