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The Net Effect on Binding Energy
of the Addition or Subtraction
of a Proton or Neutron from
Integral Alpha Particle Nuclides

The alpha particle, He4 nuclide, and other nuclides which could contain an integral number of alpha particles have a relative large binding energy compared to nuclides having slightly fewer protons or neutrons required to form alpha particles. This phenomenon is explored quantitatively in another webpage. The results were interpreted to indicate that nuclei contain substructures of alpha particles.

This webpage investigates the binding energies of nuclides which could contain an integral number of alpha particle plus or minus a proton or neutron. To make the presentation more concise the integral alpha particle nuclides will be referred to as α nuclides and the nuclides which are integral alpha particles plus or minus a proton or neutron will sometimes be referred to as α+p, α+n, α-p, and α-n nuclides.

For convenience let the binding energy function be expressed symbolically as B(Q) where Q is two dimensional vector of the numbers of protons and neutrons. Then B(α) represents the binding energy of a nuclide in which the numbers of protons and neutrons could make up an integral number of alpha particles; i.e., the numbers of protons and the number of neutrons are even numbers and equal to each other. B(α+p) is then the binding energy of a nuclide in which the number of protons is one unit greater than the number of neutrons and hence the number of protons less one combined with the number of neutrons would make up an integral number of alpha particles.

The data for the difference between the binding energy the α+p nuclides and the α+n and the corresponding α nuclides are plotted in the following graph.

What is being plotted is [B(α+p)−B(α)], the reddish line of dots, and [B(α+n)−B(α)], the blue line of dots.

What the graph indicates is that the proton and neutron have quite different effects on binding energy. The addition of a proton at most increases the binding energy by less than 3 MeV and declines to negative levels. The addition of a neutron, on the other hand, results in higher binding energies reaching the 12 MeV level for larger nuclides. The lesser effect of proton compared to a neutron is explanable in terms of the positive charge of the proton. At small separation distances the nuclear force between two protons is far greater than the electrostatic repulsion between them. Because of the effective short range of the nuclear force at greater separation distances the electrostatic repulsion is greater than the nuclear force attraction. In larger nuclides the separation distances of some protons are reaching the point where the electrostatic repulsion is more significant.

It is notable that the graph lines for an additional proton and for an additional neutron, while differing drastically in overall level and shape, do show the same local ups and downs. This would seem to be quite significant in terms of how an additional nucleon affects the nuclear structure.

If a nucleus contains substructures of alpha particles then the effect on binding energy of being short one nucleon for the formation of an alpha particle should be greater than the effect of having an extra nucleon. The data for [B(α)−B(α-n)] and [B(α)−B(α-p)] are shown in the following graph.

This stronger effect is confirmed in the case of a missing neutron. There is a greater effect for a missing proton as well.

An interesting relationship comes to light when the effect on binding energy of going from one nucleon less than an integral alpha particle nuclide to one nucleon more than an integral alpha particle nuclide. First consider the case for neutrons.

For small nuclides the completion of an alpha particle brings a big jump in binding energy, as much as 22 MeV. But for larger nuclides the effect comes down to about 18 MeV. The addition of a neutron for small nuclides is in the range of 2 to 7 MeV. However for larger nuclide the effect rises to about 14 MeV.

For a proton the pattern is more complicated. For small nuclides the completion of an alpha particle brings a large increase in binding energy, as much as 22 MeV. But for larger and larger nuclides the effect steadily declines, reaching a level of 6 MeV for the largest nuclides. The addition of a proton for smaller nuclides decreases the binding energy instead of increasing it. The effect rises to an increase of about 4 MeV for nuclides largest enough to contain 5 to 12 alpha particles.

The interesting result is the difference between the two curves in the two graphs. Those differences are

[B(α+n)−B(α)]−[B(α)−B(α−n)] = [B(α+n)−2B(α)+(B(α-n)]
[B(α+p)−B(α)]−[B(α)−B(α−p)] = [B(α+p)−2B(α)+(B(α-p)]

These quantities can be called the second differences of binding energy with respect to the neutron number and the proton number and may be designated as Δn²B and Δp²B, respectively.

The second differences are so close for neutrons and protons that it is virtually impossible to distinguish between them graphically. The difference data is plotted as red dots for protons and as the yellow covered area for neutrons.

The numerical values are:

The Second Differences for the
Effects of Protons and Neutrons
on the Binding Energies of
Integral Alpha Particle Nuclides
1 21.779527 21.46329 0.316237
2 17.44012 17.23372 0.2064
3 14.013356 13.775493 0.237863
4 11.527132 11.520412 0.00672
5 10.412158 10.103248 0.30891
6 9.42156 9.20142 0.22014
7 8.83683 8.70614 0.13069
8 6.58743 6.4008 0.18663
9 6.64809 6.4646 0.18349
10 7.2433 7.2785 -0.0352
11 7.0353 6.7697 0.2656
12 6.015 5.752 0.263
13 5.7824 5.4998 0.2826
14 6.4711 6.394 0.0771
15 4.6627 4.771 -0.1083
16 5.07 5.52 -0.45
17 5.3 5.51 -0.21
18 5.4 5.27 0.13
19 4.675 4.16 0.515
20 4.9 4.6 0.3
21 5.0 4.7 0.3
22 4.8 4.3 0.5

Rather than the second differences being equal a more accurate characterization is that

Δn²B = Δp²B − 0.17271 MeV

Another way of displaying the relationship between the second differences is in terms of a scatter diagram and the correlation between the second differences.

The regression equation for the second difference for neutrons in terms of the second difference for protons is

Δn²B = −0.10395 + 0.99252Δp²B
R² = 0.99761

The t-ratio for the regression coefficient is 91.4. The coefficient of determination R² indicates that 99.761 percent of the variation in the second difference for neutrons is explained by the variation in the second difference for protons. The correlation between the two second differences is 0.99881.

Thus although the differences between the effects of protons and neutrons on the binding energy of integral alpha particle nuclides are quite different in magnitude and their relationships with the numbers of alpha particles in the nuclides, there is a remarkably close relationship between the second differences. The interpretation and significance of this relationship remains to be determined.

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