For a doubly magic + 2 nucleons system, the 2 nucleons are in and orbitals. If there is no residual interaction, all possible -states formed by -orbital and -orbital are degenerated. As the total Hamiltonian is a simple sum of “single-particle” Hamiltonian with the mean field, that nucleons “do” not interact with each others.

Lets denote the single-particle energies for the an orbitals are and , respectively. The interaction energy between the two valence nucleons is:

where is the nuclear Hamiltonian. And is the wave function of two nucleons that coupled to spin .

Sometimes, there could be 2 configurations couples to same . for example, (d5/2)(d5/2) and (d5/2)(s1/2) can both couple to spin 2, and 3. For example, see this post. In this case, the wave function is a linear combination of the two configurations. Thus, the Hamiltonian is a bit complicated,

The solution is in this post. And the observed energy levels are ,

and the states have spectroscopic factors,

Now, the theoretical background is laid down. Experimentally, lets take the 18O , and the 17O(d,p)18O reaction as an example.

The 17O is a single d5/2 neutron on top of 16O. Adding another neutron on 17O, the new neutron will couple to that d5/2 neutron, any state contains d5/2 will be populated. We restrict ourselves only to the d5/2 and s1/2 states. For the state, we pick the ground state and the 5.34 state as the mixing between the (d5/2)(d5/2) and (s1/2)(s1/2).

The spectroscopic factor of the ground state is 1.22, and the SF of the 5.34 MeV state is 0.16. We have to normalize the SF.

Next, we have to estimate the single-particle energy. The binding energy for the a d5/2 neutron is , And the binding energy for 2 neutrons and their interaction is . Thus, the neutron interaction energy is .

Thus, we have

And

Thus, the TBME of the d5/2-d5/2 neutrons coupled to J = 0 is -3.29 MeV. The TBME of the d5/2-s1/2 neutrons coupled to J = 0 is -1.71 MeV.

The is the single-particle energy of the s1/2 neutron. The 1/2+ state next to the 5/2+ ground state of 17O is 0.87 MeV. Thus, , then, .

For the 4+ state of 18O, it can only be coupled by (d5/2)(d5/2) neutron. Teh excited energy is 3.55 MeV, The TBME of J = 4 is -0.35 MeV.

For the 2+ state, we can repeat the same method. We take the 1.98 MeV and 3.92 MeV, with spectroscopic factors for the d5/2 neutron adding are 0.83 and 0.66 respectively. The 2+ state can be formed by (d5/2)(d5/2) and (d5/2)(s1/2) configuration.

Thus,

Then, the (d5/2)(d5/2) TBME is -1.06 MeV, the TBME for (d5/2)(s1/2) is -1.71 MeV, and the off-diagonal TBME is -0.96 MeV.

At last, the 3+ state can only be formed by (d5/2)(s1/2). The excited state in 18O is 5.375 MeV. , thus, the TBME for (d5/2)(s1/2) is 0.60 MeV.

In summary,

J | T | configuration | TBMEs [MeV] | |

0 | 1 | diagonal | d5/2 – d5/2 | -3.29 |

2 | 1 | diagonal | d5/2 – d5/2 | -1.06 |

4 | 1 | diagonal | d5/2 – d5/2 | -0.35 |

0 | 1 | diagonal | s1/2 – s1/2 | -0.92 |

0 | 1 | (d5/2)(d5/2) – (s1/2)(s1/2) | -1.71 | |

2 | 1 | diagonal | (d5/2)(s1/2) – (d5/2)(s1/2) | -1.71 |

2 | 1 | (d5/2)(d5/2) – (d5/2)(s1/2) | -0.96 | |

3 | 1 | diagonal | d5/2 – s1/2 | 0.60 |

The total Hamiltonian is

We can compare these value with our previous estimation. For the (d5/2) (d5/2) configuration.

I state the previous estimation in here, with the single-particle energy already subtracted.

The 4+ state interaction action energy is -0.35 MeV. In the previous estimation, it is . which is very good agreement.

The 3+ state is actually repulsive. In the previous estimation, we ignored the mutual interaction, because the (d5/2) and (s1/2) states are eigen state and there is no mutual interaction. Possible issue mixing with d3/2 or the interaction is not a delta function.

The TBME for the (d5/2)(d5/2) state is -3.29 MeV and in previous estimation, it is , good agreement. The TBME for the (d5/2)(d5/2) – (s1/2)(s1/2) is -1.71 MeV, and the TBME for (s1/2)(s1/2) is -0.92 MeV, they agree with the previous estimation of -1.83 MeV and -1.057 MeV, respectively.

For the TBMEs, the off-diagonal term is -0.92, which is agreed with the . However, the diagonal terms are -1.06 and -0.71, which is a bit different from and . Nut we have to notice that the 2+ state is offset in previous calculation.