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An Estimation and Analysis of the Binding Energy
Involved in the Formation of Neutron-Neutron Spin Pairs

Most nuclei have a mass deficit; i.e., their masses are less than the mass of the neutrons and protons of which they are composed. That mass deficit expressed in units of energy is known as the binding energy. The difference between the binding energy of a nuclide and that of one containing one less neutron is called the incremental binding energy of the neutron (IBEn). The incremental binding energy of a proton is defined similarly. The incremental binding energy (IBE) of a nucleon is composed of two parts.

One part is due to the interaction of the additional nucleon with the other nucleons through the strong force. The other part is due to the formation of nucleon pairs. Those pairs can be neutron-neutron, proton-proton and/or neutron-proton. This second part can be estimated by a procedure illustrated below for additional neutrons.

The second method largely reproduces what results from the first method. Therefore in the following analysis it is only the first method and the nuclides with an even number of neutrons that are used. Also if the number of neutrons is less than the number of protons then an additional neutron will form a neutron-proton pair as well as a neutron-neutron pair. Therefore only the nuclides in which the number of neutrons is greater than or equal to the number of protons are used in the analysis.

Note that in the case of Neodymium, shown above, when the number of neutrons reaches 82 there is a sharp drop and a change in the amplitude of the the odd-even fluctuations. The sharp drop comes after 82 because at 82 a neutron shell is filled. Any additional neutron goes into a higher shell.

Below are shown a selection of cases for the heavier nuclides.

For these a linear function of the number of neutrons in the shell would be appropriate. A comparison of the last three cases is notable.

The lower level for the odd 91 case reflects some pairing phenomena involving the protons.

The cases for some middle level nuclides are shown below.

These are generally regular. The sawtooth pattern for Chromium (p=91) may reflect the formation of alpha particles.

For the lighter nuclides there are regularity but the patterns differ more drastically for the different proton numbers.

The disparaties are shown below.

For heavier nuclides and limiting the comparison to those with an even number of protons the match is quite remarkable. For example, consider the case of Lead (p=82), Polonium (p=84) and Radon (p=86).

There is a very close match for the shell containing 83 to 126 neutrons (seventh shell), but divergence for the next shell.

The Frequency Distribution of the Values of the
Pair Enhancements for the Formation of a Neutron Pair

The notion of a pair formation involving just the two nucleons is not sustainable, but there might be a clustering of values. This possibility is examined by constructing a histogram (frequency distribution) for the values.

There is a degree of clustering in the range of 1.0 MeV upto 1.1 Mev, but the values extent of a range from about 0 to 4 Mev. The five values beyond 3 MeV are not shown in the histogram. However, most of the values are in the range from 0.5 MeV to 1.7 MeV.

Statistical Approximation of the Relationship
Between the IBEn for a Nuclide and Its
Number of Neutrons and Protons

The ultimate goal of the analysis is to find an equation in terms of the numbers of neutrons and protons that most closely approximates the incremental binding energies of the nuclides. As a step in that direction consider fitting curves to particular cases. The graphs for Lead and Aluminum display relatively smooth ogee shapes; i.e., lines in which the curvature reverses.


The relevant variable is m the number of neutrons in the shell. The regression equation is

IBEn = c0 + c1m + c2m² + c3

For Aluminum the results are

IBEn = 2.67887 − 0.62904m + 0.11020m² − 0.00615m³
[-14.4] [10.2] [-8.6]
R² = 0.99662

The numbers in the square brackets [] below a coefficient is its t-ratio, the ratio of its value to its standard deviation. A t-ratio with a magnitude of 2 or greater is statistically significant at the 95 percent level of confidence. The values found are highly significant statistically.

For Lead using the values below 126 the results are

IBEn = 3.62166 − 0.21715m + 0.007174m² − 8.756×10-5
[-3.3] [3.1] [-3.4]
R² = 0.99007

The coefficients for the two cases have the same signs but differ drastically in magnitude. There is the possibility that the relevant variable is not the number of neutrons in the shell but the proportion of the shell which is filled, m divided by the shell capacity.

Let r be the ratio of m to the shell capacity, 14 in the case of Aluminum and 44 for Lead.

The results for Aluminum are

IBEn = 2.67887 − 8.80657r + 21.59883r² − 16.8762r³
[-14.4] [10.2] [-8.6]
R² = 0.99662

For Lead they are

IBEn = 3.62166 − 9.55473r + 13.88877r² − 7.45856r³
[-3.3] [3.1] [-3.4]
R² = 0.99007

The values of the coefficients differ but they are on the same order of magnitude.

The data for Tungsten displays the ogee shape but the curve is not smooth.

The regression equation is

IBEn = 1.01791 + 4.37128r − 15.69316r² + 13.20728r³
[4.6] [-6.1] [6.4]
R² = 0.96093

Sodium is a interesting special case.

The graph for 14 neutrons and above only has one curvature. The regression equation for those data points is

IBEn = 1.71209 − 0.7523r − 1.75425r² − 4.52660r³
[-0.3] [-0.2] [-0.4]
R² = 0.99330

The case of Radon is interesting and part of the graph has some similarity with that of Sodium.

The data points for n less than 126 are for the seventh shell, those above 126 for the eighth shell.

The regression equation for the incremental binding energies of neutrons in the isotopes of Radon for the seventh shell is

IBEn = −14.80495 + 56.49635r − 64.70619r² + 23.66222r³
[12.8] [-11.9] [10.7]
R² = 0.99982

For the eighth shell the results are

IBEn = 0.70514 + 6.83594r − 46.1498r² + 95.1848r³
[6.7] [-5.5] [4.8]
R² = 0.9658

The data for Platinum display an irregular curve and the degree to which a cubic regression equation fits them is notably lower.

The regression equation is

IBEn = 2.6098 − 7.41432r + 12.5769r² − 7.03439r³
[-2.6] [2.4] [-2.3]
R² = 0.70171

Although the shape display for the data for Platinum is irregular it is the same shape as the the data for the nearby nuclide of Osmium (p=76).

The match is more striking if the curve for Platinum (p=78) is shifted backwards by two units.

(To be continued.)

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