San José State University
Thayer Watkins
Silicon Valley
& Tornado Alley

The Number of Interactions and the
The Incremental Structural Binding Energies
of Alpha Particles in Nuclides

This webpage investigates an implication of a model of nuclei in which neutrons and protons combine wherever possible into alpha particles. The binding energy of a nuclide is thus composed of two components. One component is the binding energy of the alpha particle substructures and the other is the binding energy due to the potential energy involved in putting these substructures together into an arrangement. This binding energy will be referred to as structural binding energy (SBE).

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 principal quantum number 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 the binding energies and structural binding energies are given in the Alpha Particle Substructure of Nuclei. The values of the incremental structural binding energies are tabulated below.

#Alphas SBE
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

The model implies that when an alpha particle is added its increment in structural binding energy comes from its interactions with the other alpha particles. The incremental structural binding energy per interaction should be the same for the alpha particles added in the same shell. The number of interactions for the n-th alpha particle is just (n-1). When an alpha particle is added to a higher shell the increment per interaction should be smaller because in a higher shell the distances to the other alpha particles are greater.

Here is the graph of the incremental structural binding energies per interaction for the nuclides containing an integral number of alpha particles.

In the diagram the average incremental structural binding energy per interaction in each shell is plotted along with values for each alpha particle. The first shell contains only one alpha particle and there is no structural binding energy. The second shell contains two particles but the datum for only the third alpha particle is available. The third shell consists of the fourth through seventh alpha particle. The fourth shell consists of the eighth through fourteenth alpha particle and the fifth the fifteenth through twenty fifth. The results generally confirms the implication that the incremental structural binding energy per interaction is constant for the particles in each shell. The confirmation is strongest for the fifth shell and less solid for the fourth and lower shells. Note that the average of the values for the fourth and fifth alpha particles is much closer to the average for their shell than the individual values are.

The ratios of the averages for the shells are of some interest because they are inversely related to the radii of the shells. The ratio of the average for the fourth shell to that of the fifth is 5.018; that of the third to to that of the fourth is 2.438 and that of the second to that of the third 2.076.

The data from which the previous graph was constructed are given below.

Alphas ISBE per
Shell Average
3 3.683272 3.683272
4 2.387311333 1.774274829
5 1.18246225 1.774274829
6 1.8632714 1.774274829
7 1.664054333 1.774274829
8 0.992585143 0.727624939
9 0.82992075 0.727624939
10 0.782291778 0.727624939
11 0.5127026 0.727624939
12 0.699238727 0.727624939
13 0.6616105 0.727624939
14 0.615025077 0.727624939
15 0.193451857 0.145003651
16 0.1774884 0.145003651
17 0.134645375 0.145003651<
18 0.141430941 0.145003651
19 0.150240333 0.145003651
20 0.179175053 0.145003651
21 0.1402163 0.145003651
22 0.104967905 0.145003651
23 0.109287545 0.145003651
24 0.13062287 0.145003651
25 0.133513583 0.145003651

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