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Evidence Concerning Alpha Particle Structures in Nuclei

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.

The Alpha Nuclides

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.

plus One
less One
Effect of
Adding one
Effect of
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
19 638.1 649.74 622.3 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

Subtracting One Neutron from Each Alpha Nuclide

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.

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

BE = α + β/Nγ
or, equivalently
ln(BE-α) = ln(β) − γln(N)

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

ln(BE−13.1835) = 2.21398 − 0.29773ln(N)
R² = 0.99076

Thus the results indicate a value of 13.1835 MeV associated with the completion of an alpha particle and a shell-related phenomenon which is smaller for the larger shells.

Adding One Neutron to Each Alpha Nuclide

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.

Comparison of the Effects

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

DBE = α + β/Nγ
or, equivalently
ln(BE-α) = ln(β) − γln(N)

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

ln(DBE+3.45) = 3.50124 − 0.38705ln(N)
R² = 0.97143

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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