The purpose of this webpage is to derive estimates of the increments to binding energy due to the interaction of neutrons with protons in various nuclear shells. The methodology for this is derived from the following propositions.

• 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_n(Δ_pBE) = 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.

• 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 are considered.

  Shell Number	1	2	3	4	5	6	7	8
  Capacity	2	4	8	14	22	32	44	58
  Range	0 to 2	3 to 6	7 to 14	15 to 28	29 to 50	51 to 82	83 to 126	127 to 184

• 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 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.

More details on the relationships for the various shells are given in Interaction Binding Energies of Neutrons.

Based upon the previous displays the appropriate regression equation for the data is

Δ_nBE(N, P) = c₀ + c₁P + c₂e(P) + c₃d(P≥N) + c₄N + c₅e(N)

where N and P are the numbers of neutrons and protons, respectively. The variable e(P) is 1 if P is even and zero otherwise and likewise for e(N). The variable d(P>N) equal 1 if P≥N and zero otherwise.

The data for the incremental binding energies of neutrons were partitioned into sets in which all of the neutron numbers and all of the proton numbers were in particular shells. The results of the regressions are given below.

Where 0 appears for the coefficient of d(P≥N) it means that there was no variation in d(P≥N) for that data set. Either P≥N for all of the cases or none of the cases in the data set.

The regression coefficients for P represent the interaction binding energy of neutrons and protons in various shells. This information can be arranged in matrix form, as below.

The pattern of the values is very regular. For example, consider the interaction binding energies of neutrons and protons with the same shell numbers.

In terms of the logarithms of the interaction binding energies the pattern is even more regular.

The Incremental Binding Energies of Protons

Now the question is whether the previous results are supported by the data on the incremental binding energies of protons.

The coefficient of N, in this case, gives the interaction binding energy for the last neutron and the last proton. The results can be displayed in the same matrix form as was used for the results based on the incremental binding energies of neutrons. Since the equality is of interest it is worthwhile to display the two sets of results in the same table. In the table below the results for neutrons are the upper number and the results for protons the lower number.

The correspondence of the two measurements of the interaction binding energies of neutrons and protons is demonstrated by their scatter diagram.

The correlation of the two measures is 0.978. The regression of interaction binding energy based upon the incremental binding energies of protons, IE_p, upon IE_n gives the following results:

IE_p = 0.016538 + 0.98939IE_n

R² = 0.95693

The regression coefficients are not significantly different from 0 and 1. Thus there is no systematic bias in either of the estimates of the interaction binding energies of neutrons and protons. Both are equally good so the best estimate of values of the interaction binding energies are their averages. These average values are displayed in matrix form.

The interaction binding energy of a neutron with a proton having the same shell number declines sharply with shell number. It is essentially an exponential decline, as shown by the plot of the logarithm of the interaction binding energy versus shell number.

More generally, the interaction binding energies decline with increasing shell number in any direction.

Conclusions

The proposition that the two cross differences of binding energy both measure the interaction binding energies of the last neutron with the last proton is born out. Furthermore that interaction binding energy is a function only of the shells of the last neutron and last proton. The interaction binding energy of a neutron and proton is a systematically regular function of the shell numbers involved and hence is likely to be explainable.