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An Analysis of the Changes in the Relationship between the Binding Energy
Involved in the Formation of Proton-Proton Spin Pairs and the Number
of Protons in the Nuclide as the Number of Neutrons in the Nuclide Varies

Most nuclei have a mass deficit; i.e., their masses are less than the mass of the protons and neutrons 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 proton is called the incremental binding energy of the proton (IBEp). The incremental binding energy of a neutron is defined similarly. The incremental binding energy (IBE) of a nucleon is composed of two parts.

Note that in the case shown above when the number of protons reaches 20 and 28 there is a moderately sharper drop and a change in the amplitude of the the odd-even fluctuations. The sharp drop comes after these numbers of protons because a shell or subshell is filled. Any additional proton goes into a higher shell. There is also a sharper drop when the number protons reaches the level of the number of neutrons. Up to that point each time a proton is added there is a proton-neutron pair formed.

One part of the incremental binding energy 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 proton-proton, neutron-neutron and/or proton-neutron. Later such effects will be referred to as the enhancement of binding energy due the formation of such nucleon pairs. This second part can be estimated by a procedure illustrated below for additional protons.

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 protons that are used. Also if the number of protons is less than the number of neutrons then an additional proton will form a proton-neutron pair as well as a proton-proton pair. Therefore only the nuclides in which the number of protons is greater than or equal to the number of neutrons are used in the analysis.

The restriction imposed of considering only the nuclides in which the number of protons exceeds the number of neutrons severely limits the data. There are only 141 nuclides meeting that requirement. Limiting the cases considered to the nuclides with an odd number of protons further limits the data. There are at most three data points for each number of neutrons. The pattern of the relationship between the binding energy due to proton pair formation and the number of protons varies. Because the effect varies with the character of the nuclide, in this case the number of protons in the nuclide, there is something more involved than simply the formation of a proton-proton pair. Apparently there is some rearrangement of the other nucleons that occurs when the proton pair is formed.

There appear to similarities in the relationships. These similarities are illustrated in the graph below.

However in the case of other sequences of neutron numbers there are very little in the way of similarities.

A Search for Structure

An analysis of the data for binding energy enhancement due to neutron pair formation it was concluded that the operative variable is the proportion of the neutron shell which is filled. The level, slope and the shape of the relationships varied as a function of the number of protons in the nuclides. For protons there are at most three data points for each case. There are three data points for the nuclides with an even number of neutrons in the nuclide for 8 through 26. Only for 19 among the nuclides with an odd number of neutrons are there three data points. The slopes of the relationships were computed using linear regression. For the cases with two data points the slopes were computed as the ratio of the change in the binding energy enhancement to the change in the number of protons (2). The graph of the result is shown below.

There is no statistically significant slope to the relationship and the average of the twenty cases is −0.01467 MeV per proton.

A similar construction was carried out using the second differences as a measure of the curvature of the relationship. This shown below.


The enhancement of binding energy due to the formation of a proton spin pair varies with the number of protons and neutrons in the nuclice in which the pair is formed. The is generally a gradual change in the relationship between the pair enhancement of binding energy and the number of protons but the variation is more limited with a proton shell.

(To be continued.)

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