Nuclei are held together by the forces due to spin pairing and the nuclear strong force but the electrostatic force plays a role in the repulsion between protons. The spin pairing between nucleons, neutron-neutron, proton-proton and neutron-proton, is in all cases effectively a force of attraction. It is a different matter for the strong force between nucleons. As will be illustrated later neutrons repel neutrons and protons repel protons, not only electrostatically, but through the strong force as well. The strong force between a neutron and a proton is an attraction. Nuclei are held together by the attraction between neutrons and protons and by the spin pairing forces.

Spin pairing is an exclusive matter. One neutron can form a spin pair with one other neutron and with one proton but no other. The same applies for one proton. The binding energies associated with pair formation is on the order of several million electron volts (MeV). The nuclear strong force between two nucleons is far smaller. But the strong force is not exclusive. Therefore in nuclei with a larger number of nucleons the large number of interactions through the strong force can be dominant compared to the force associated with spin pair formation. However the strong force on a nucleon is the net effect of the repulsion of like nucleons and the attraction of unlike nucleons. The conventional assertion is that all nucleons attract each other through the strong force. But this is because conventional practise in physics assesses the nuclear forces by scattering experiments. In scattering only a few nucleons are involved and that is the situation in which the force associated with spin pair formation is dominant. The effect of the strong force is manifested only in terms of the binding energies of larger nuclei.

First Differences and Second Differences of Binding Energies

Consider a nucleus with N neutrons and P protons. The binding energy of the nucleus is the sum of the binding energies due to the interactions of the nucleons. This involves the neutrons with other neutrons, the protons with other protons and the neutrons with protons. Let n and m be indices for the neutrons and F_nm be the binding energy due to the interaction of the n-th neutron with the m-th neutron, G_pq the binding energy due to the interaction of the p-th proton with the q-th proton and H_np the binding energy due to the interaction of the n-th neutron with the p-th proton. To avoid double counting m is always greater than n. Likewise let p and q be indices for the protons with q greater than p. The binding energy of a nucleus with N neutrons and P protons is then

BE(N, P) = ΣΣ_n<mF_nm + ΣΣ_p<qG_pq + ΣΣH_np

The summation for n is from 1 to N and for m from n+1 to N. Likewise the summation for p is from 1 to P and for q from p+1 to P.

The arrangement is depicted visually below. Both the white and colored squares represent binding energies due to nucleon interactions.

Now consider subtracting BE(N-1, P) from BE(N, P). BE(N-1, P) is represented by the colored squares in the display above. All of the interactions of protons with other protons are obviously eliminated. Also the interactions of the protons with the first N-1 neutrons are eliminated. And the interactions among the first N-1 neutrons are eliminated. The result is Δ_N(N, P) = ΣF_nN + ΣH_Np

where the summations above are from 1 to N-1 for n and from 1 to P for p. This is the first difference or increamental binding energy.

The arrangement is depicted visually above as the squares in white.

Consider now two subtractions from Δ_N(N, P). First consider the subtraction of Δ_N(N-1, P). This produces

Δ_NN(N, P) = F_N,(N-1)

This is the interaction of the last neutron with the next-to-last neutron. Such a quantity is called a second difference.

Next consider the subtraction of Δ_N(N, P-1) from Δ_N(N, P). The result of this subtraction is

Δ_N,P(N, P) = H_N,P(N, P)

This is the interaction of the last neutron with the last proton. This is in the nature of a second difference but usually this quantity is called a cross difference.

The same procedure applies with taking the first differences with respect to the number of protons. An interesting thing is that there are two ways of computing the binding energy due to the interaction of the last neutron and the last proton.

Empirical Values of First and Second Differences

Consider the isotopes of Tin. The number of protons in the tin isotopes is 50.

The sawtooth pattern is evidence of the formation of neutron pairs. The sharp drop at 82 neutrons is the effect of the filling of a neutron shell.

Abstracting away the effect of the pair formation the second differences are negative, thus indicating from the above analysis that the force between the last and next-to-the-last neutrons is repulsion.

On the other hand, a comparison of the first differences for P=50 with P=49, shown below, shows there is a positive value for the cross difference, thus indicating the binding energy due to the interaction of the last neutron with the last proton is positive and hence the force between them is an attraction.

The first value of -0.6 MeV should not be included because it represents a shift between shells. The average difference over the shell is 0.388 MeV. This the average interaction binding energy of the last neutron and the last proton.

This is only for P=50 but the same results apply for the other proton numbers.

The Incremental Binding Energies of Protons

The comparison for N=49 and N=50 is below.

The average difference is 0.465 MeV. This is an alternate estimate of the average interaction binding energy of the last neutron and the last proton. Not all cases are available for protons that exist for neutrons. If the average for the neutron case is computed over the same range as for the protons the average is 0.399 MeV. The correspondence of the two alternate measures of the binding energy due to the interaction of the last neutron and the last proton is reasonable. The positive values indicate an attraction between a neutron and a proton.

The downward slope of the incremental binding energy as a function of the number of protons indicate that the interaction binding energy of the last proton with the next-to-last proton is negative, indicating a repulsion through the strong force.