The purpose of this webpage is to derive estimates of the increments to binding energy due to the interaction of neutrons with neutrons and protons 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. They are:

  Δ²_n,nBE = Δ_n(Δ_nBE) = Δ_nBE(n, p) − Δ_nBE(n-1, p)
  and
  Δ²_{p, p}BE = Δ_p(Δ_pBE) = Δ_pBE(n, p) − Δ_pBE(n, p-1)

  The case is made elsewhere that the second difference Δ²_n,nBE is equal to the interaction binding energy of the last neutron with the next to last neutron and Δ²_{p, p}BE is the interaction binding energy of the last proton with the next to 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 33 through 37 protons and tabulated as a function of the number of neutrons contained in them. The graph of the results is as follows.

  

  The sharp drop the level after number of neutrons equals 50 represents the effect of the filling of a neutron shell. The sharp drop after 36 neutrons is a different phenomenon. This change in the pattern after the number of neutrons equals the number of protons has to do with the non-formation of neutron-proton pairs. This will be referred to as the n=p phenomenon. The sawtooth pattern is due to the effect on binding energy of the formation of neutron-neutron pairs. The slope of the line corresponds to the second difference Δ²_{n, n}. The fact that the relationship is linear over the range of the fifth neutron shell indicates that the interaction energy of a neutron with the previous neutron in the shell is constant over the range of the 29th through 50th neutrons.

  An appropriate regression equation for the data would be

  Δ_nBE(n, 36) = c₀ + c₁n + c₂d(n>36) + c₃e(n)

  where e(n) is 1 if n is even and zero otherwise. The variable d(n>36) equal 1 if n>36 and zero otherwise. This would apply for n≤50.

  The results of the regression are

  Δ_nBE(n, 36) = 23.85583 − 0.30390n − 2.0691d(n>36) + 3.17100e(n)

  The coefficient of determination (R²) for this regression equation is 0.9965. This means that 99.65 percent of the variation in Δ_n(n, 36) is explained by the three variables, n, d(n>36) and e(n).

  Thus the estimate of the interaction binding energy of the 35th through 50th neutrons with of the previous neutron in the fifth neutron shell is −0.30390 MeV. Its negativity indicates a repulsion between those neutrons. The regression coefficients are all statistically significantly different from zero at the 95 percent level of confidence.

• When the data for the elements with p=34 through p=37 are plotted in the same graph there is a striking degree of parallelism. The graph includes all of the cases for which there are values for Δ_n over the full range of the fifth neutron shell.

  

  Each case has the n=p phenomenon and the sawtooth pattern but other than these 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 fifth neutron shell with the previous neutron in the fifth neutron shell 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(N>P) + 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(N>P) equal 1 if N>P 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 N represent the interaction binding energy of a neutron with the previous neutron in the shell.

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 same statistical analysis was carried out for the incremental binding energies of protons. The results are displayed in the same sort of table that was used for the analysis of neutrons. However the roles of N and P are reversed in the table.

The coefficient of P, in this case, gives the interaction binding energy for neutron and the previous proton.

Conclusions

The proposition that the second differences of the binding energy of neutrons measure the interaction binding energies of the last neutron with the next to last neutron is born out. Furthermore that interaction binding energy is a function only of the shell of the neutron. The same applies to the second difference of the binding energy of protons.