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The Correspondences of the Incremental
Binding Energy Relationships
of Neutrons and Protons

Recent studies (protons and neutrons) examined the relationship between incremental binding energies of the neutrons and the protons and the number of nucleons of the same type in the nuclide. This was to look for evidence of subshells within the neutron shells. Such evidence was found. The numbers indicating the sizes of the subshells agreed for neutrons and protons. What was not known was the correspondence of the relationship. Were the changes in the patterns the same for the two types of nucleons? This study examines such correspondences.

Here is the display of the relationships in the range which includes the magic number 50.

It is notable that the slopes of the relationships are negative and that that of the proton is more negative than that of the neutron. In a previous study it was found that the slope of such relationships is equal to the binding energy effect due to the interaction of the last nucleon with the next-to-last nucleon of the same type. The fact that the slopes are negative indicates that the interaction of a nucleon with a nucleon of the same type is a repulsion.

The effect of interaction on binding energy can be represented in term of nucleonic charges for the proton and the neutron. If two particles have nucleonic charges of Q1 and Q1 then the force between them is proportional to Q1Q2 and likewise the potential energy and binding energy due to their interaction.

Let the nucleonic charge of a proton be designated as 1 and that of neutron as q. The interaction of two neutrons is then proportional to q². Two protons are subject to electrostatic repulsion so the charge of the proton is effectively (1+δ) where δ is the ratio of the force due to electrostatic repulsion to the repulsion due to the nucleonic force. The value of δ varies with the separation distance of two protons, being smaller for small separation distances and larger with larger separations.

If the electrostatic repulsion between protons is ignored the ratio of the slope of the neutron relation to that of the proton relation should be q². Over the range from 40 to 50 the incremental binding energy of a neutron decreases by 3.493 MeV whereas that for the proton decreases by 8.290 MeV. The ratio of those two figures, which should be at least approximately equal to q², is 0.42135. If this is q² then q is equal to −0.649. This value is remarkably close to the value of −2/3 found in other studies.

The data for a range that includes the magic number 82 is:

The incremental binding energy of neutrons decreaes by 1.797 MeV over the range of 70 to 82. The incremental binding energy of protons decreaes by 6.420 MeV over the same range. The ratio is 0.2799 and its square root is −0.5291. This can be considered to be equal to q/(1+δ). A value of 0.25 for δ would give a value of 0.6613 for q.

The data for p=20 and n=25 display some of the same characteristics as the two displays above but the sharp drops for 20 and 25 and the differing slopes of the two relationships make it difficult to determine if the changes in pattern are the same for both relationships.

It is notable that the incremental binding energy of neutrons decreases by 1.329 MeV over the interval 18 to 20. The incremental binding energy of protons decreases by 2.810 MeV over the same interval. The ratio is 0.473 and its square root is 0.6876, an estimate of the nucleonic charge of a neutron relative to that of a proton.

(To be continued.)

For more on nuclear subshells see Subshells.

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