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There is considerable evidence that alpha particle exist as substructures of nuclei. One test of this hypothesis is to compare the effect of adding one more neutron to a nuclide which could contain only alpha particles (hereafter called alpha nuclides) with the effect of taking away one neutron. If taking away one neutron destroys an alpha particle then there would be large drop in the binding energy. The conventional estimate of the binding energy of the alpha particle (He 4 nuclide) is 28.3 million electron volts (MeV) whereas adding a neutron would increase the binding energy by amount based upon the interaction of the neutron with the other nucleons of the nucleus.
There are 25 nuclides which could contain an integral number of alpha particles. The conventional estimates of their binding energies are shown in the table below with the conventional binding energies of the nuclides which are alpha nuclides plus one neutron, and those which are the alpha nuclides less one neutron.
#α  Alpha Nuclides  Alpha Nuclides plus One Neutron  Alpha Nuclides less One Neutron  Effect of Adding one Neutron  Effect of Subtracting one Neutron 
1  28.295674  27.41  7.718058  0.885674  20.577616 
2  56.49951  58.1649  37.6004  1.66539  18.89911 
3  92.161728  97.108063  73.4399  4.946335  18.721828 
4  127.619336  131.76266  111.9556  4.143324  15.663736 
5  160.644859  167.40597  143.7805  6.761111  16.864359 
6  198.25689  205.58756  181.7248  7.33067  16.53209 
7  236.53689  245.01044  219.3572  8.47355  17.17969 
8  271.78066  280.42222  256.7383  8.64156  15.04236 
9  306.7157  315.5046  291.4622  8.7889  15.2535 
10  342.052  350.4147  326.4108  8.3627  15.6412 
11  375.4747  385.0047  359.175  9.53  16.2997 
12  411.462  422.044  395.128  10.582  16.334 
13  447.697  458.3802  431.514  10.6832  16.183 
14  483.988  494.235  467.347  10.247  16.641 
15  514.992  525.223  499.99  10.231  15.002 
16  545.95  556.01  530.37  10.06  15.58 
17  576.4  586.62  560.67  10.22  15.73 
18  607.1  617.93  591  10.83  16.1 
19  638.1  649.7  4 622.3  11.64  15.8 
20  669.8  681.3  653.7  11.5  16.1 
21  700.9  712.3  684.8  11.4  16.1 
22  731.4  743.4  715.1  12  16.3 
23  762.1  774.3  745.6  12.2  16.5 
24  793.4  806  12.6  
25  824.9  835.6  10.7 
The effect of subtracting one neutron is not necessarily equal to the binding energy of an alpha particle. The two protons and one neutron have their own effects on binding energy. Their effects are however relatively smaller than that of creating an alpha particle. The graph of the effect of subtracting one neutron is shown below.
The effects are plotted versus the number of neutrons in the alpha nuclide in order to demonstrate the involvement of shell phenomena. The effect drops more sharply at the numbers in set {2, 6, 14, 28}. These are magic numbers, not the conventional magic numbers but the modified magic numbers which are associated with a particular algorithm. Generally the level of the effects is constant with slight deviations from constancy within each shell. For the last three shells the slopes are positive but less positive the higher the shell. This is a pattern that is found elsewhere in the physics of nuclei. Only for the 4 to 6 shell is the slope slightly negative.
The regression equation parameters for the data within the shells are given below.
Neutron Shell  Slope  Intercept 
8to14  0.21077965  14.2413926 
16to28  0.131209286  13.02693286 
30to46  0.0741  13.09664444 
If only the values at {2, 6, 14, 28} are considered there is an inverse relationship. This suggests functional relationship of the form
where N is the number of neutrons.
When α is chosen to maximize the coefficient of determination (R²) of the regression equation the resulting equation is
Thus the results indicate a value of 13.1835 MeV associated with the completion of an alpha particle and a shellrelated phenomenon which is smaller for the larger shells.
Below is shown the effects on binding energy of adding one neutron to the alpha nuclides.
In this display the effect of adding one neutron to the null alpha nuclide is included. The null alpha nuclide is just the case of zero alpha particles. In this case the local minima occur at magic numbers, but two of these magic numbers, 8 and 20, are conventional magic numbers which are not included in the modified set of magic numbers. They happen to be equal to the sum of the two previous magic numbers; i.e., 8=2+6 and 20=14+6. The number 42 happens to also be the sum of the two previous magic numbers;; i.e., 42=28+14. Minima also occur for the magic numbers 2 and 50.
The level is generally increasing except for variations within the shells. This can be attributed to the increasing number of interactions an additional neutron is involved in. But the slope of this general relationship is decreasing with increasing number of neutrons because in higher level shells the distance between an additional neutron and the other particles is larger.
Now the effects of subtracting one neutron from each alpha nuclide can be compared with the effects of adding one neutron.
The relationships seem to be approaching a constant separation. This is illustrated in the graph of the differences of the two relationships.
The patterns are so different for the two effects that it seems to indicate a mutual exclusivity; i.e., if a neutron is involved in the formation of an alpha particle then it is not involved in the interactions with the other nucleons in the nucleus. This would be a very important phenomenon if true.
It is notable that the relationships for the smaller nuclides appears to be more regular than for the larger nuclides. Again the shape suggests a relationship of the form
where DBE is the difference in the two effects on binding energy and again N is the number of neutrons.
When α is chosen to maximize the coefficient of determination (R²) of the regression equation the resulting equation is
Surprisingly the method indicates that asymptotically the effect of an additional neutron would be greater than the effect of the formation of an alpha particle. The display of ln(DBE+3.45) versu ln(N) is shown below.
The implication the above material is that alpha particles are formed within nuclides only up to a certain limit. Thereafter it is more effective energetically for the nucleons to remain separate.
The pattern of the effect of subtracting one neutron from each alpha nuclide indicates that there is a roughly constant effect on binding energy of the formation of an alpha particle with in a nucleus. There is a relatively small effect having to do with the interaction of the neutron which completes the formation of the alpha particle with the other nucleons in the nucleus.
(To be continued.)
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