Nucleon-pair wave functions in a single- j shell

In this work we study approximate and exact solutions for nucleons in a single-j shell, from the perspective of nucleon-pair basis states, i.e., those coupled by pairs with good spins. We find that for four, five, and six particles in the 0h11/2 shell, a selected set of independent nucleon-pair basis states leads to approximate solutions of a realistic two-body interaction, without resorting to the diagonalization. We analytically show that for six particles in the j=112 shell, two nucleon-pair states with J=3 and 11 - which are coupled by three pairs of spin 0, 2, and 4 and by pairs of spin 0, 2, and 10, respectively - are eigenstates of any two-body interactions. In particular, we construct general analytic expressions for states of definite seniority numbers ν=3,4,5 in terms of nucleon-pair basis states, based on which we further derive exact wave functions for a few eigenstates of any two-body interactions in the midshells of j=72,92,112. The exact wave functions given here should be useful in interpreting electromagnetic moment and transition properties of corresponding nuclei.