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to the Interaction of Neutron Pairs |
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There is a theorem that says that the second difference in binding energies for a particle gives the interaction binding energy of the last one of those particles with the next to last one. In order to keep the results from being affected by the odd-even fluctuations due to spin pairs only nuclides with even numbers of neutrons and protons will be considered. There are 728 such nuclei and for 628 the incremental binding energy of a neutron pair can be calculated.
Let nn and pp be the numbers of neutron and proton pairs, respectively. Let BE(nn, pp) be the binding energy of the nuclide with nn neutron and pp proton pairs. The incremental binding energy of a neutron pair is then
The second difference in binding energy with respect to the number of neutron pairs is then
For the second difference to measure the interaction of the last two neutron pairs, nn, nn-1 and nn-2 must be in the same neutron shell.
The analysis begins with the first differences in the binding energies of the isotopes of heavier elements with an even number of neutrons. There is more regularity for such elements than for the lighter elements. First take the case of the isotopes of Plutonium shown below..
The relationship appears to be linear and the variation in the number of neutron spin pairs explains 99.7 percent of the variation in the incremental binding energy. The slope of the relationship, based upon regression analysis, is −-0.61273 MeV per neutron pair per neutron pair. This is 0.15323 MeV per neutron-neutron interaction.
However despite the apparent linearity of the above relationship the second differences reveal a pattern of deviations from linearity.
This relationship appears to be approximately sinusoidal with a period of 6. Such a function explains 89.1 percent of the variation in the second differences. There is a slight upward trend in the values. Including nn as an explanatory variable raises the coefficient of determination (R²) to 0.95147.
The integral of a sinusoidal function is a sinsusoidal function so a sinusoidal function may be included in the regression equation for the incremental binding energies of neutron pairs. When this is carried out the coefficient of determination (R²) for the regression is raised from 0.9880 to 0.9999.
The sinusoidal deviations from linearity are not always an element of the relationship between incremental binding energy and the number of neutron pairs. Take the case of Polonium, p=84.
The relationship appears to be two lines separated with a drop after nn=63, which corresponds to a filled shell at 126 neutrons. The second differences are not sinusoidal.
The relationship of incremental binding energies and the number of neutron pairs is very regular and almost a straight line.
The slope is negative and become increasingly negative.
For Neodymium there are two shells involved divided by nn=41, which corresponds to the filled shell magic number of 82.
For Cadmium the slope is negative but become increasingly less so as the number of neutron pairs increases.
For Krypton there are two lines separated by the nn=25, which corresponds to the filled shell magic number of 50.
For Chromium (p=24) the relationship is a negative sloping line but less regular than previous examples.
For Magnesium (p=12), as for the case of Chromium, the relationship is a negative sloping line but less regular than previous examples.
(To be continued.) .
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