The binding energy of a nucleus represents the amount of energy which would be required to break it up into its constituent nucleons (neutrons and protons). The binding energy of a nucleus is a function of the numbers of neutrons and protons which make it up, BE(n, p). The incremental binding energies (ΔBE) can be computed according to the following definitions

Δ_proton(n, p) = BE(n, p) − BE(n, p-1)
Δ_neutron(n, p) = BE(n, p) − BE(n-1, p)

Hereafter Δ_proton and Δ_neutron will be denoted as Δ_p and Δ_n.

The incremental binding energies are the first differences in BE. From them the second differences can be defined. In particular, the cross differences are defined as:

Δ²_p,nBE = Δ_p(Δ_nBE) = Δ_nBE(n, p) − Δ_nBE(n, p-1)
and
Δ²_{n, p}BE = Δ_pBE(n, p) − Δ_pBE(n-1, p)

The case is made elsewhere that the cross differences Δ²_p,nBE and Δ²_{n, p}BE are both equal to the interaction energy of the last neutron and last proton added to the nucleus. Thus the two cross differences should be equal.

The interaction binding energy of a neutron and proton may depend upon which nuclear shells they are in. Here are the nuclear shells considered in the following analysis. No possible subshells within the shells were considered.

The Incremental Binding Energies
of the Neutrons in Nuclides as a Function
of the Number of Protons They Contain

The IBE_n's were computed for those nuclides containing 24 neutrons and tabulated as a function of the number of protons contained in them. The graph of the results is as follows.

There is a jump in the level after the number of protons equals the number of neutrons. This change in the pattern after the number of proton equals the number of neutrons will be referred to as the p=n phenomenon. The slope of the line corresponds to the cross difference Δ_{p, n}. The fact that the relationship is linear indicates that the interaction energy of the neutrons and protons are constant over the range of the 15th to 28th protons.

A regression equation for the data is

Δ_nBE(24, p) = c₀ + c₁p + c₂d(p≥24) + c₃e(p)

where e(p) is 1 if p is even and zero otherwise. The variable d(p>24) equal 1 if p≥24 and zero otherwise.

The results of the regression are

Δ_nBE(24, p) = -6.415372857 + 0.846427143p
+ 1.69936d(p≥24) + 0.645178571e(p)

The coefficient of determination (R²) for this regression equation is 0.9986. This means that 99.86 percent of the variation in Δ_n(24, p) is explained by the three variables, p, d(p≥24) and e(p).

Thus the estimate of the interaction binding energy of the 24th neutron with each of the protons in the fourth proton shell is 0.84643 MeV. The regression coefficients are all statistically significantly different from zero at the 95 percent level of confidence.

When the data for all of the cases for which there are values for Δ_n over the full range of the fourth proton shell are plotted in the same graph there is a striking degree of parallelism.

Each case has the p=n phenomenon but other than that the plots are linear and have essentially the same slope. What this means is that the interaction energies of each of the neutrons in the fourth neutron shell with each of the protons in the fourth proton shells are all essentially the same. The above estimate of the value is 0.84643 MeV.

Now the question is whether this is verified by the data on the incremental binding energies of protons.

The Incremental Binding Energies
of the Protons in Nuclides as a Function
of the Number of Neutrons They Contain

The data for the nuclides with 16 protons illustrates that the same phenomena occurs for the Δ_p as occurs for Δ_n.

The regression result is

Δ_pBE(n, 16) = -9.509898901 + 1.042886593n
+ 1.305123846d(n≥16) + 0.47218e(n)

The coefficient of determination (R²) for this regression equation is 0.9984.

The estimates of the interaction energies of neutrons and protons in the fourth shells, 0.84643 MeV and 1.04289 MeV, are statistically significantly different. However, as such things go, the two values are remarkably close. Their average, 0.94466 MeV, is the best estimate so far of the interaction energies of neutrons and protons in the fourth shells.

The above suggests that the binding energy of the interaction of neutrons and protons is a function of the neutron and proton shell numbers; i.e., IE_n,p(S_n,S_p) where S_n and S_p are the neutron and proton shell numbers, respectively. So then it is known that IE_n,p(4,4)=0.94466 MeV.

Hereafter in order to make the tasks manageable only the data and results based upon the incremental binding energies of neutrons will be considered.

The Fifth Shells

Both the fifth shell for neutrons and the fifth shell for protons contain the 29th through 50th nucleons. The display for the case of interaction between neutrons in their fifth shell and protons in their fifth shell is shown below. This is based upon the incremental binding energies of neutrons.

The graph lines are essentially linear and parallel. The slopes of the lines are all about 0.5 MeV per particle.

The pattern based upon the incremental binding energies of protons is as follows.

It is the same pattern as the graph based upon the incremental binding energies of neutrons; i.e., linear except for a slight odd-even fluctuation and a jump after the number of neutrons equals the number of protons. However in this case the slopes of the lines are about 0.6 MeV.

The Sixth Shells

The capacities of the sixth shells are both 32. Tabulating all of the data for display of the relationships is tedious, so only the first nine data points are used for the following display.

The indication is that here too the relations are essentially linear and parallel. There is a slight odd-even fluctuation about a line.

The Seventh Shell

A display illustrating the relationships for the seventh shell is

For this case the relationships are more strictly linear but there is slight difference in the slope for n=127 compared to the other numbers of neutrons. The slope of the line for n=124 neutrons is 0.15 MeV.

The Eighth Shell of Neutrons and the Seventh Shell of Protons

There are no stable nuclides for which the proton number reaches the eighth shell. The data for the cases for the interaction of the neutrons in the eighth shell with protons in the seventh shell are as follows.

The slope of the line for n=156 neutrons is 0.22 MeV.

The Third Shells

Generally the relationships are more regular for the larger nuclides than for the smaller ones. The data for the interaction of neutrons and protons in their third shells looks irregular but much of the irregularity is due to the n=p effect.

The regression equation for n=10 is

Δ_n(10, p) = -2.2883465 + 1.2808535p
+ 5.409303d(p≥10) + 0.4850555e(p)

So the interaction between neutrons and protons in their third shells is about 1.28 MeV. The coefficient of determination (R²) for this regression equation is 0.98781. Thus the statistical fit is good.

The Second Shells

For the case of interactions between nucleons in the second shells there is more irregularity than with previous cases. The relationships are less clearly linear and the slopes change with the number of neutrons.

There are only four data points but regression equation for this case is

Δ_n(5, p) = -6.165085 + 1.95945875p
−0.90714125e(p)

The coefficient of determination for this regression equation is 0.98362, so the fit is reasonably good.

The Matrix of Interaction Energies
of Neutrons and Protons by Shells

The evidence so far indicates that a matrix of the following form can be completed, at least for interactions for the third shell and above.

The interaction between neutrons and protons in adjacent shells can be computed but that task is left for a later work.