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The Regularities and Irregularities of the
Binding Energies of Alpha Module Nuclides

Neutrons and protons form spin pairs whenever possible, but this pairing is exclusive in the sense that one neutron can form a spin pair with only one other neutron and with only one proton. The same applies to protons. Thus chains of the form -p-n-n-p-, or equivalently, -n-p-p-n- are created. These units can be called alpha modules because they roughly correspond to alpha particles. The chains may close to form rings. The simplest such ring is the alpha particle.

The nuclides that a composed entirely of alpha modules are called alpha module nuclides. Here their binding energies of the investigated for indications of filled shells. Here are those binding energies along with other relevant data.

0 0
1 28.295674 28.295674
2 56.49951 28.203836
3 92.161728 35.662218
4 127.619336 35.457608
5 160.644859 33.025523
6 198.25689 37.612031
7 236.53689 38.28
8 271.78066 35.24377
9 306.7157 34.93504
10 342.052 35.3363
11 375.4747 33.4227
12 411.462 35.9873
13 447.6 36.138
14 483.988 36.388
15 514.992 31.004
16 545.95 30.958
17 576.4 30.45
18 607.1 30.7
19 638.1 31
20 669.8 31.7
21 700.9 31.1
22 731.4 30.5
23 762.1 30.7
24 793.4 31.3
25 824.9 31.5

When the binding energies are plotted versus the number of alpha modules there is no hint of irregularities.

There is not a hint of any irregularity in the relationship. The statistical regression of binding energy BE on the number of alpha modules #α is

BE = 33.50526#α + 0.57388
[157.0] [0.2]

The coefficient of determination for this equation is 0.9990. This might seem to be good but it is not. The standard error of the estimate is 8.16 MeV. With an average binding energy of 419.4 MeV for the data this corresponds to a coefficient of variation of 0.0195. The t-ratio for the coefficient of #α is 157.0, which indicates that it is greatly different from zero statistically, but the t-ratio for the constant term is only 0.2. This indicates that the constant term is not significantly different from zero at the 95 percent lellvel of confidence.

If the constant is eliminated from the regression the result is

BE = 33.53902#α

The variations in the relationship between binding energy and the number of alpha modules show up in terms of the incremental binding energies of the alpha modules, IBE where

IBE(#α) = BE(#α) − BE(#α−1)

The plot of IBE versus #α reveals some interesting patterns.

There is definitely a shell that extends from above 14 to 25 alpha modules. These correspond to there being more than 28 neutrons and 28 protons. The first shell seems to have a capacity of two alpha modules. This corresponds to 4 neutrons and 4 protons. In the conventional theory the first nucleon shells have a capacity of 2 and the next shell a capacity of six and thus a filled shell (magic) number of 8. My previous work suggests that six is the magic number for the second shell.

The conventional theory puts the next magic number of nucleons at 20, corresponding to 10 alpha modules. There is something that occurs at 10 alpha modules but the above graph indicates that the more relevant limit is at 7 alpha modules, which corresponds to 14 nucleons.

The relation of binding energies to the number of alpha modules above 14 (corresponding to 28 of each nucleon) that the regression equation for this range of the data has a coefficient of determination of 0.999990. A bent-line regression equation may be fitted to the entire data set using bend points at 2#α and 14#α. The regression result is

BE = 28.21203 #α + 7.38967 u(#α-2)
− 4.62202u(#α-14) + 0.02788
[54.2] [13.4]
[-40.7] [0.03]

where u(z) stands for the ramp function such that u(z) = 0 for z<0 and u(z)=z for z≥0.

The coefficient of determination of this equation is 0.999987 and the standard error of the estimate is 0.97 MeV. This corresponds to a coefficient of variation for the error in the equation of 0.23 of 1 percent.

The constant in the above equation is not significantly different from zero at the 95 percent level of confidence. when the constant term is eliminated from the regression the results are

BE = 28.2268 #α + 7.37472 u(#α-2)
− 4.62178u(#α-14)
[121.7] [26.0] [-41.7]

The coefficient of determination and so forth are essentially the same as for the previous regression. The nice part of this regression is that the coefficient of #α, which should be essentially the same as the binding energy of an alpha particle, is close; 28.22680 MeV compared to 28.29567 MeV.


The binding energies of the alpha module nuclides clearly define a shell extending from above 14 alpha modules to 25. The first shell apparently is filled with two alpha modules. Above two alpha modules the evidence is not clear; somethings apparantly happen at 4, 7 and 10 alpha modules but not so definitely as to define shells.

The statistical performance of a model that has filled shells at 2 and 14 is excellent. The coefficient of determination of the regression is 0.99999 and the coefficient of variation of the difference between the actual value and the regression prediction is 0.23 of 1 percent.

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