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on the Structural Binding Energies of Alpha Particles |
This is an analysis of nuclear binding energies based upon the neutrons and protons of a nucleus forming alpha particles whenever possible. This means that the binding energy of a nuclide is made up of one part due to the formation of the alpha particles and another, called the structural binding energy, which is due to the arrangement of the alpha particles in the nucleus. The structural binding energy for a nuclide can be computed by deducting from the binding energy the binding energy due to the formation of alpha particle, which is simply the number of alpha particles times 28.29457 million electron volts (MeV), the binding energy of one alpha particle. This computed structural binding energy is positive for all 2931 nuclides except for four: Carbon 8, Boron 7, Berylium 6 and Berylium 5. Berylium 5 is the one nuclide out of 2931 which has negative binding energy.
The nuclides which could contain an integral number of alpha particles are an interesting starting point. The graph of the structural binding energies of these nuclides as a function of the number of alpha particles in the nuclide displays an interesting shape.
This form indicates a shell structure for the alpha particles of the nuclei. The graph for those nuclides which could contain an integral number of alpha particles plus one neutron has a similar shape. The data for the alpha nuclides and the alpha plus one neutron nuclides are shown below.
The sharp drop at 25 alpha particles may be due to the proximity to 50 neutrons.
From the above graph it is clear that the effect of an additional neutron depends upon the number of the alpha particles in the nuclide. To see the relationship more clearly the differences are plotted below.
The sharp drop at 25 alpha particles may be because of the proximity to the magic number of 50 newaf
The statistical explanation for this pattern is in terms of the nuclear magic numbers. The magic numbers for neutrons are {2, 6, 14, 28, 50, 82, 126}. The numbers 8 and 20 are also magic but in a different class from the others. The magic numbers for neutrons and protons correspond to the set {1, 3, 7, 25, 41, 63} for alpha particles. The differences in structural binding energies is then a bent line with bend points at 3 and 7. The regression equation of this form has a coefficient of determination (R²) of 0.9690 and a standard error of the estimate of 0.629 MeV. The regression coefficients are all statistically significant at the 95 percent level of confidence.
The comparison between the data and the regression estimates is shown in the graph below.
The slopes are such that for 1 to 3 alpha particles the increase in structural binding energy is 2.55 MeV per additional alpha particle. From 3 to 7 alpha particles the increase is 1.10 MeV per additional alpha particle and from 7 to 25 it is 0.205 MeV per additional alpha particle.
The graph of the differences in the structural binding energies of the alpha+1proton and the alpha nuclides is shown below.
The comparison of the data for the differences and the bent line regression estimate are shown below.
The coefficient of determination (R²) for this regression equation is 0.900 and the standard error of the estimate is 0.465 MeV.
For 1 to 3 alpha particles the structural binding energy increases 1.67 MeV per additional alpha particle. For 3 to 7 alpha particles the increase is 0.28 MeV per alpha particle and for 7 to 25 alpha particles it is −0.25 MeV per alpha particle.
(To be continued.)
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