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The Quantification of the Interaction Energy Between
Alpha Particles and Neutron Pairs Within Nuclei

This is an examination of the binding energies of nuclides which could contain an integral number of alpha particles plus a fixed number of neutron pairs. It is based upon the presumption that within a nucleus the neutrons and protons form alpha particles whenever possible. The formation of an alpha particle entails a binding energy of 28.295 million electron volts (MeV). The binding energy from the formation of alpha particles accounts for most of the binding energies of nuclides. If BE is the binding energy of a nuclide and α is the number of alpha particles it contains then the excess binding energy XSBE is given by

XSBE = BE − 28.295*α

The difference in the excess binding energies of the alpha+n_pairs of neutrons nuclides and the alpha+(n-1)_pairs of neutrons is the incremental binding energy of the n-th neutron pair. The graph of the incremental excess binding energy of the first neutron pair is shown below.

This case exhibits a shell-type structure in which the incremental excess binding energy shifts after a shell is filled. The neutron shells are filled at a set of numbers called magic numbers. The traditional magic numbers are {2, 8, 20, 28, 50, 82, 126}. There is also evidence that 6 and 14 are magic numbers. The two graphs above are displayed to test whether the crucial variable is the number of alpha particles in the nuclide or the number of neutrons. Clear the number of neutrons is the relevant variable for determining changes in the pattern of the incremental excess binding energy.

There is a special significance to the plot of the incremental excess binding energy due to an additional neutron pair and the number of alpha particles in the nuclide. The increments in excess binding energy with respect to increases in the number of alpha particles is the interaction energy between the last alpha particle and the last neutron pair. This is approximately constant within a shell. Since the number of neutrons is just equal to twice the number alpha particles plus two it is easy to go back and forth between the different representations.

A piecewise linear function can be fitted to the data using the two functions u(z) and d(z) where

u(z) = z if z≥0 and u(z)=0 otherwise
d(z) = 1 if z≥0 and u(z)=0 otherwise

The plot has a drop after the critical numbers and then a rise. Therefore the regression equation used is of the form

IXSBE = Σ(ciu(n-(ni+1)) + bid(n-(ni+1)) + aid(n-(ni+2)))

The critical levels of the neutron numbers are {8, 20, 28, 50}. A comparison of the data values for the first effect of the first neutron pair and the regression estimates of those data values is shown below.

The fit is so close that the estimates virtually overlay the actual values. It is a little easier to see the minor discrepancies between the regression estimates and the actual values if a smooth curve is draw through the points of the two sets of data, as shown below. The coefficient of determination (R²) for the regression is 0.99710.

The graph for the effect of the second neutron pair is shown below.

The levels and slopes for the shells are obtained as the cumulative sums of the corresponding regression coefficients.

In the case of the third neutron pair a new phenomenon develops, the number 14 as a magic number.

For the fourth neutron pair the nature of the smaller shells is uncertain, but the shells beyond 28 and 50 are clearly defined.

And likewise for the fifth neutron pair.

For the sixth neutron pair the magic number of 82 enters into the pictue.

For the seventh neutron pair there appears to be subshells for the (51-82) shell .

Likewise for the eighth neutron pair

And also for the ninth and tenth neutron pair

The pattern continues. For example, for the thirteenth neutron pair is

In due course a critical point appears at magic number 126; e.g. for the eightteenth neutron pair.

In this case as with others there is evidence of subshells within the larger shell.

The continuation of the pattern is seen in the case of the twenty first neutron pair.

The pattern continues going beyond the range of magic number 126 until, as a complete surprise, a critical point occurs for the twenty fifth neutron pair at number 152.

The number 152 has not been known to be a nuclear magic number, but there is a breakpoint at 152 for the remaining cases up to the final one for the twenty ninth neutron pair.

(To be continued.)

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