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Function of their LImited Range Intra- and Intershell Interactions |
The alpha nuclides are those nuclides which could be composed entirely of alpha particles. There is evidence that whenever possible the nucleons in a nucleus form alpha particles. The structural binding energy of a nuclide is its binding energy less the binding energy that is accounted for by the formation of alpha particles. This structural binding energy of alpha nuclides has a relatively simple pattern as a function of the number of alpha particles in the nuclides.
This pattern indicates something in the nature of shells. There is abundant evidence of a shell structure of nuclides which is usually expressed in term of magic numbers. The magic numbers represent filled shells. The conventional magic numbers are 2, 8, 20, 28, 50, 82 and 126. Elsewhere it is proven that 6 and 14 are also magic numbers and furthermore that 8 and 20 are different from the other magic numbers. The corrected sequence of magic numbers is 2, 6, 14, 28, 50, 82 and 126. This means that the capacities of the shells are 2, 4, 8, 14, 22, 32 and 44 nucleons. For the alpha nuclides this means capacities of 1, 2, 4, 7 and 11 because there are only 25 alpha nuclides.
For each alpha nuclide the number of alpha particles in each shell was determined. For example, for the alpha nuclide with 8 alpha particles there is one particle in the first shell, two in the second and five in the third. From this information the number of interactions between and within shells was determined. For the alpha nuclide with 8 there could be two interactions between the particles in the first and second shells and five between those in the first and third shells. The two particles in the second shell could have one interaction with each other and ten with the five particles in the third shell. The five particles in the third shell could have ten interactions; i.e., 5*4/2.
A previous study carried out a statistical analysis of the structural binding energies of the alpha nuclides based upon the above formulation.
When the data for the interactions were used to try to explain the variation in the structural binding energies of the alpha nuclides it was found that 99.98 percent of the variation was explained by the linear regression equation. However some of the coefficients were not statistically significant at the 95 percent level of confidence. These included the constant term and the interaction of the one particle in the first shell with the particles in the other shells. They also included the interactions of particles separated by more than one shell. Thus there was no significant interaction of the particles in the second shell with the particles in the fourth and fifth shell. Likewise there was no significant interaction of the particles in the third shell with those in the fifth shell. Furthermore the interactions of particles within the fifth shell were not significant. When these interactions were eliminated from the regression equation the coefficient of determination (R²) was 99.56 percent.
Here are the regression coefficients and their t-ratios (ratio of coefficients to their standard deviations). An interaction of (n,m) is the interaction of the alpha particles in the n-th shell with those in the m-th shell.
Interaction | Coefficient | t-Ratio |
(2,2) | 7.78909 | 14.0 |
(2,3) | 2.66119 | 10.5 |
(3,3) | 1.62201 | 5.4 |
(3,4) | 1.56283 | 23.4 |
(4,4) | .23523 | 3.0 |
(4,5) | 0.38386 | 56.4 |
With a correlation of 0.998 between the regression estimates and the data the statistical performance is impressive. But perhaps significant result is that the only significant interactions are between alpha particles in adjacent shells. Alpha particles have two sides; a neutron side and a proton side. It is very reasonable that the significant interactions would be between the neutrons of the alpha particles in one shell with the protons of the particles in the next shell.
The result that alpha particles only interact with other alpha particles in the same or adjacent shells suggests that the intertactions can take place over a limited distance. Thus each alpha particle can interact with its nearest neighbors or nearest and next nearest neighbors. This would mean that the number of interactions is just proportional to the number of alpha particles in a shell. Thus the number of interactions of n particles in one shell and m particles in an adjacent shell would be equal to the minimum of n and m.
The regression equation including all of the variables but no constant is as follows:
Interaction | 5,5 | 4,5 | 4,4 | 3,5 | 3,4 | 3,3 | 2,5 | 2,4 | 2,3 | 2,2 | 1,5 | 1,4 | 1,3 | 1,2 |
Coefficient | 2.66873 | 0.2292 | 1.53532 | -0.614 | 2.56488 | 9.65035 | 0.3352 | 3.00754 | -5.03182 | 7.36655 | -0.42139 | 1.48701 | 2.54341 | -7.45838 |
t-Ratio | 15.1 | 0.6 | 29.9 | -1.1 | 5.5 | 22.6 | 0.3 | 2.7 | -4.6 | 8.6 | -0.3 | 1.0 | 1.7 | -5.5 |
The coefficient of determination (R²) for the equation is 0.99738.
When the variables whose t-ratio is less than 2 and thus whose coefficients are not significantly different from zero at the 95 percent level of condifence are eliminated the regression results are as follows.
Interaction | 5,5 | 4,4 | 3,4 | 3,3 | 2,4 | 2,3 | 2,2 | 1,2 |
Coefficient | 2.64595 | 1.49329 | 2.62198 | 9.56569 | 3.73356 | -3.73532 | 7.81041 | -7.90224 |
t-Ratio | 51.8 | 37.1 | 5.8 | 23.5 | 5.0 | -5.1 | 9.1 | -5.7 |
The coefficient of determination (R²) for the equation is 99.56 percent.
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