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The Incremental Structural Binding Energies
of Alpha Particles in Nuclides

The masses of nuclides are less than the sum of the masses of the protons and neutrons which they contain. This mass deficit translated into energy equivalence is called the binding energy of the nuclide. The binding energy of a nuclide could be composed of two components. One component could be the binding energy of substructures which are composed of protons and neutrons and another the binding energy due to the potential energy involved in putting these substructures together into an arrangement. Previous work indicates that the protons and neutrons within a nucleus form alpha particles whenever possible. Any additional protons and neutrons beyond the alpha particles substructures form pairs; neutron-neutron, neutron-proton and proton-proton. These pair formations are not mutually exclusive; i.e., a neutron may form a pair with a proton as well as with another neutron.

A major component of the binding energy of a nuclide would then be that due to the formation of alpha particles and nucleon pairs. The rest of the binding energy would be due to the configuration of those alpha particles and nucleon pairs and any excess protons or neutrons. This binding energy will be referred to as structural binding energy (SBE).

There are accepted values of binding energies for alpha particles (28.3 MeV) and for neutron-proton pairs (2.2 MeV) but no accepted value for neutron pairs.

The structural binding energy for those nuclides that could contain an integral number of alpha particles has an interesting form.

The above graph suggests that there are shell structures of the alpha particles within the nuclei. A shell is a collection of particles with the same quantum number(s) and hence at the same distance from the center of the nucleus. There is no significant increase in binding energy for two alpha particles but for three there is. The additional structural binding energy for the number of alpha particles above two is roughly constant at about 7.3 MeV per additional alpha particle until a level of 14 alpha particles is reached. Thereafter the increase is about 2.7 MeV per additional alpha particle, as shown below.

The numerical stability of the increments in structural binding energy can be examined by computing the increase in binding energy as the number of potential alpha particles increases. This is the incremental structural binding energy (ISBE) of an alpha particle.

The numerical values are important for further investigations.

The Binding Energies of Nuclei Which Could
Contain an Integral Number of Alpha Particles
Element Neutrons Protons Binding
Number of
Alpha Particles
of Alpha Particles
He 2 2 28.295674 1 28.295674 0
Be 4 4 56.49951 2 56.591348 −0.091838
C 6 6 92.161728 3 84.887022 7.274706
O 8 8 127.619336 4 113.182696 14.43664
Ne 10 10 160.644859 5 141.47837 19.166489
Mg 12 12 198.25689 6 169.774044 28.482846
Si 14 14 236.53689 7 198.069718 38.467172
S 16 16 271.78066 8 226.365392 45.415268
Ar 18 18 306.7157 9 254.661066 52.054634
Ca 20 20 342.052 10 282.95674 59.09526
Ti 22 22 375.4747 11 311.2524 64.22229
Cr 24 24 411.462 12 339.548088 71.913912
Fe 26 26 447.697 13 367.843762 79.853238
Ni 28 28 483.988 14 396.139436 87.848564
Zn 30 30 514.992 15 424.43511 90.55689
Ge 32 32 545.95 16 452.730784 93.219216
Se 34 34 576.4 17 481.026458 95.373542
Kr 36 36 607.1 18 509.322132 97.777868
Sr 38 38 638.1 19 537.617806 100.482194
Zr 40 40 669.8 20 565.91348 103.88652
Mo 42 42 700.9 21 594.209154 106.690846
Ru 44 44 731.4 22 622.504828 108.895172
Pd 46 46 762.1 23 650.800502 111.299498
Cd 48 48 793.4 24 679.096176 114.303824
Sn 50 50 824.9 25 707.39185 117.50815

As is shown in the table the binding energy of a nuclide which could contain multiple alpha particles is in excess of the binding energies of the alpha particles it might contain. This difference, which will be called the structural binding energy, has the advantage over the straight binding energy in that it is not dependent upon the estimated mass of the neutron. Its value depends only upon the measured masses of the charged nuclei.

#Alphas SBE ISBE
1 0 0
2 -0.091838 -0.091838
3 7.274706 7.366544
4 14.43664 7.161934
5 19.166489 4.729849
6 28.482846 9.316357
7 38.467172 9.984326
8 45.415268 6.948096
9 52.054634 6.639366
10 59.09526 7.040626
11 64.22229 5.12703
12 71.913912 7.691622
13 79.853238 7.939326
14 87.848564 7.995326
15 90.55689 2.708326
16 93.219216 2.662326
17 95.373542 2.154326
18 97.777868 2.404326
19 100.482194 2.704326
20 103.88652 3.404326
21 106.690846 2.804326
22 108.895172 2.204326
23 111.299498 2.404326
24 114.303824 3.004326
25 117.50815 3.204326

Thus for a nuclide that could contain two to fourteen alpha particles the binding energy increases by 35.6 MeV for each additional alpha particle. This 35.6 MeV comes from the 28.3 MeV of the alpha particle itself plus about 7.3 MeV for the effect of the additonal alpha particle on the arrangement within the nucleus. It is notable that the figure of 7.3 MeV for an additional alpha particle is close to the 7.1 MeV figure for the average binding energy per nucleon in the alpha particle.

For nuclides containing fourteen to twenty five alpha partices the effect on an additional alpha particle on binding energy is an increase of 28.3 MeV for the alpha particle itself plus about 2.70 Mev for the effect of the additonal alpha particle on the arrangement within the nucleus.

The breakpoint comes at 56Ni. It has been long known that there is something special about Iron, Nickel and Cobalt nuclides. It is notable that the range over which the increment due to an additional alpha particle is about 7.3 MeV is from 2 to 14, a span of 12 alpha partices. The range for which the increment is about 2.7 MeV is from 14 to 25, a span of 11. There does not exist an integral alpha particle nuclide beyond 25. It appears that there is some arrangement of alpha particles to which an additional one may be added up to a total of twelve. This could be characterized as a shell of alpha particles. Beyond twelve there is a different shell to which alpha particles may be added.

The Case for the Nuclides Which Could Contain
an Integral Number of Alpha Particles
Plus Multiple Excess Neutrons

This graph of the incremental structural binding energies of alpha particles is the key to the spatial arrangement of nuclei. To see that the fluctuations are not just random consider the the same sort of graph for nuclides which a made up of an integral number of alpha particles plus one neutron.

Here is a graph with the previous two graphs superimposed.

The pattern is essentially the same for the two cases. There is clearly a different shell for more than 14 alpha particles. The sharp drop in ISBE occurs after 14 alpha particles for both cases. For less than 14 alpha particles the pattern is more irregular but the relative maxima and relative minima occur at the same or nearly the same values of the numbers of alpha particles.

To see the further persistence of the pattern for nuclides containing more excess neutrons the following graph was constructed which includes the data for nuclides containing an integral number of alpha particles plus two, three and four excess neutrons.

Because no accepted figure is available for the binding energy of neutron pairs the structural binding energies are computed based only upon the binding energy in excess of that involved in the formation of the alpha particles.

Again the shell for more than 14 alpha particles persists. However beyond about 25 alpha particles there is another new shell. The sharp drops in ISBE occur at a smaller number of alpha particles for the cases with a larger number of excess neutrons.

The Case for the Nuclides Which Could Contain
an Integral Number of Alpha Particles
Plus Multiple Excess Protons

The graph of the data for the alpha plus one proton (α+1p) nuclides is shown below.

It looks generally like the graph for the alpha plus one neutron (α+1n) case and the superimposition of the graphs reveals that the patterns are close.

The incremental structural binding energies for the α+1n case are all higher than the corresponding α+1p case. Below the numerical differences are displayed.

There seems to be a cyclic pattern with a cycle length of five or six alpha particles.

The persistence of the pattern of incremental structural binding energies is further demonstrated by displaying the three cases of the α, the α+1n and the α+1p nuclides all in the same diagram.

Clearly there is a pattern and the pattern is modified by the presence of an excess neutron or an excess proton. However, the incremental structural binding energy of an alpha particle is the sum of the incremental binding energies of two neutrons and two protons. Thus the pattern of the relationship of the ISBE for alpha particles could be merely that of that for neutrons and protons. On the other hand the pattern for neutrons and protons could be merely a consequence of the pattern of ISBE for alpha particles.

(To be continued.)

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