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

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.

The pattern of the relationship between the binding energy due to neutron pair formation and the number of neutrons varies. Some such as for Lead, Aluminum and Tungsten, seen below, the relationship is like an ogee curve.



Others have a nondescript irregular shape, such as those for Thorium, Protactinium and Uranium, shown below.



Although the pattern is irregular there is a notable closeness when plotted together. A comparison of the last three cases is notable.

The lower level for the odd 91 case reflects some pairing phenomena involving the protons. It is therefore advisable in establishing some continuity of the pattern to compare only the nuclides with an even number of protons.

In order to test the proportion that that the shape of the relationhips varies more or less continuously with the proton number it is necessary to establish some quantative measures of shape. This can be done by fitting a cubic equation to the data.

The regressions are based upon the proportion of the neutron shell which is filled. For example, the fifth neutron shell covers the case of 29 through 50 neutrons. The capacity is 22 neutrons. If a nuclide has 40 neutrons then there are 12 neutrons in the fifth shell and the proportion filled is 12/22. The use of such proportions instead of just the number of neutrons in the shell does not affect the statistical fit but it does affect the magnitudes of the regression coefficients. Thus the regression equation used is

PEn = C0 + C1r + C2r² + C3

where r is the proportion of the neutron shell which is filled. The proportions of the neutron shell which are filled reflect the densities of the neutrons in the shell and thus the separation distances between them. The pair enhancement for neutrons (PEn) is a function of the number of protons and therefore the coefficients are effectively functions of the number of protons. The coefficients of the cubic regression equation are then the parameters of the level, slope, curvature and shape of the relationship.

The regressions were carried out for the cases of the fifth through the eighth shells.

The Results for the Seventh Neutron Shell

The constant in the regression equation establishes the level of the relationship.

The level appears to be roughly constant until the proton number reaches about 76 and then changes significantly as the proton number is increased. At 84 the level changes drastically.

The coefficient of r indicates the slope of the relationship between the binding energy due to neutron pair formation and the number of neutrons. For the seventh shell the relationship between that coefficient and the number of protons is

The coefficient of r² indicates the curvature of the relationship between the binding energy due to neutron pair formation and the number of neutrons. For the seventh shell the relationship between that coefficient and the number of protons is

The coefficient of r³ indicates a rate of change of the curvature of the relationship between the binding energy due to neutron pair formation and the number of neutrons. For the seventh shell the relationship between that coefficient and the number of protons is

The Results for the Sixth Neutron Shell

The sixth shell covers the cases involving the 51st through the 82nd neutrons.




The Results for the Fifth Neutron Shell

The fifth shell covers the cases involving the 29th through the 50th neutrons.




The Results for the Eighth Neutron Shell

The eighth shell would cover the cases involving the 127th through the 184th neutrons, but there are only five cases for which the data is available. This makes the results much less interesting geometrically than for the sixth and seventh shells.




A Comparison of the Results for Different Neutron Shells

The relationships of the regression coefficients for the different neutron shells as a function of the proton number are generally similar. A closer comparison can be made if the variable is taken to be the proportion of the proton shell that is filled. For example, if the proton number is 30 then it is in the fifth shell that extends from 29 protons to 50 protons. The proportion q is then (p-28)/22. Here are the graphs for the regression constant for the cases in the seventh and sixth neutron shells.

   

The two are not exactly the same but the similarity is remarkable.

The graphs for the slope terms of the regression equations for the seventh and sixth neutron shells show a similar general shape.

   

Likewise for the case of the curvature and cubic terms of the regression equations for the seventh and sixth neutron shells.

   

   

Conclusions

The enhancement of binding energy due to the formation of a neutron spin pair varies with the number of neutrons and protons 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 neutrons but the variation is more limited with a neutron shell. The operative variables appears to be the proportions of the nucleon shells which are filled. This variable indicates the density of the nucleons in a shell and thus the separation distances between them. Differences in the spatial scale of the shells would affect the relationship between the effect of pair formation on binding energies.

(To be continued.)


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