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 Statistical Estimates of the Interaction Binding Energies of Neutrons with Neutrons and Protons with Protons

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

#### Δproton(n, p) = BE(n, p) − BE(n, p-1) Δneutron(n, p) = BE(n, p) − BE(n-1, p)

Hereafter Δproton and Δneutron will be denoted as Δp and Δn.

• The incremental binding energies are the first differences in BE. From them the second differences can be defined. They are:

#### Δ²n,nBE = Δn(ΔnBE) = ΔnBE(n, p) − ΔnBE(n-1, p) and Δ²p, pBE = Δp(ΔpBE) = ΔpBE(n, p) − ΔpBE(n, p-1)

The case is made elsewhere that the second difference Δ²n,nBE is equal to the interaction binding energy of the last neutron with the next to last neutron and Δ²p, pBE 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.

 ShellNumber 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 IBEn'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 Δ²n, 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

#### ΔnBE(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

#### ΔnBE(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 Δn(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 Δn 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

#### ΔnBE(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.

The Regression Equation Results for the Incremental Binding Energies of Neutrons in Various Shells as a Function of the Number of Neutrons and Protons
Neutron
Shell
Number
Proton
Shell
Number
Constant
(MeV)
P
(MeV)
e(P)
(MeV)
d(P≥N)
(MeV)
N
(MeV)
e(N)
(MeV)
4510.839820.65935-0.10170 0-0.559972.659000.98373
555.296750.482140.020492.1583-0.320962.632960.9798
650.560930.41939-0.019980-0.207862.068650.97313
757.1485700.305710-0.061.254290.92345
666.150820.2980.008260 -0.201342.218520.97626
761.863930.29000 -0.007100-0.156581.725170.97026
86-8.862650.21937-0.046320-0.040701.334030.99110
3410.730530.933390.158100-0.80464.6504 0.97940
446.152248930.751460.0708732.54022-0.561273.10121 0.97270
542.150200.68663-0.023060-0.389422.17326 0.96051
235.55532.21881-0.004340-1.168816.43827 0.98815
334.004491.217720.175484.12186-0.87970.96625
431.268031.144480.066390-0.609932.3985 0.88805
223.076002.32211.382884.99037-2.337998.94596 0.92193
32-1.302011.66479-0.036850-0.60993 2.77570 0.88051
779.063690.18118-0.015520-0.140811.57530 0.94902
872.466580.228260.00102770-0.128201.33960 0.91938

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 Regression Equation Results for the Incremental Binding Energies of Protons in Various Shells as a Function of the Number of Neutrons and Protons
Neutron
Shell
Number
Proton
Shell
Number
Constant
(MeV)
N
(MeV)
e(N)
(MeV)
d(P≥N)
(MeV)
P
(MeV)
e(P)
(MeV)
5415.337270.66140.002900 -1.083273.01353 0.96599
557.529630.55628-0.03131-1.64371-0.711252.53724 0.98716
6514.522400.394240.065260-0.690402.65958 0.98976
7614.104000.27870.039770-0.5253 2.006870.98591
7712.090020.19607-0.037370-0.397081.96627 0.97660
8711.528830.257650.0219400 -0.473521.67387 0.97432
4316.108281.032160.17652 -1.72723-1.72723 4.680600.90126
448.214520.96481-0.05981-1.3428-1.11582.94595 0.98066
458.892360.59585-0.029810-0.852992.50624 0.96942
327.804131.399120-0.805725.7504 0.97358
338.180321.540630.18163-2.98841-1.629334.47439 0.96823
345.286370.88169-0.028360-0.994692.37122 0.94190
224.553872.614420.52651-5.21684-1.984587.11596 0.91530
233.548341.98532-0.047420-1.87876 3.63840.98178
666.474050.35983-0.111030-0.506482.38749 0.98479

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.