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The Effect on Binding Energy of the Addition of
Four-Nucleon Combinations to Alpha Nuclides

Nuclei are thought of as being composed of protons and neutrons. Their binding energies are computed in terms of the deficit of their masses compared to the masses of the constituent protons and neutrons. However there is a good deal of evidence that the protons and neutrons form alpha particles whenever possible.l (The nuclides which could and probably do contain an integral number of alpha particles are here called alpha nuclides.) Any additional neutrons beyond those tied up in alpha particles form pairs where possible. Therefore the binding energy of a nuclide is largely a matter of the binding energies of the subparticles (alpha particles and nucleon pairs) of which it is composed. There is an additional component of binding energy for a nuclide which comes from the arrangement of those subparticles in the nuclide. This binding energy will be referred to as excess binding energy.

When the excess binding energies are computed for the alpha nuclides and the values plotted versus the number of alpha particles in the nuclide the result is as shown below.

For the data and further detail of the alpha particle substructure of nuclides see Alpha Particle Substructure of Nuclides.

The above graph indicates a shell structure for the alpha nuclides. There is a shell of two alpha particles, then a shell of twelve and then a shell of at least eleven.

If the excess binding energies are computed for the nuclides which could contain an integral number of alpha particles plus four additional neutrons the result also shows a shell structure.

This indicates that the capacity of the third shell of alpha particles is indeed eleven.

The excess binding energy computed above took into account only the binding energy of the alpha particle substructures. The four neutrons would have binding energy of at least two neutron pairs. There is the possibility that four neutrons might form a structure of greater binding energy than that of two neutrons pairs just as the binding energy of an alpha particle is much greater than the binding energy of two proton-neutron pairs (deuterons). To investigate this possibility and others the binding energies of the alpha-plus-four-neutrons nuclides are complied along with those which contain a proton-neutron pair and another neutron pair. For comparison the effect of the addition of another alpha particle is included. These are shown below.

Binding Energies (MeV) of Nuclides Containing an Integral Number of Alpha Particles Plus a Four Nucleon Combination
Number of
alpha particles
alpha plus
4 neutrons
alpha plus
alpha +
p-n pair
+ n-n pair
1 28.295674 31.408 56.49951 28.82
2 56.49951 68.65 92.161728 79.5752
3 92.161728 110.753 127.619336 117.981
4 127.619336 151.3707 160.644859 154.40267
5 160.644859 191.836 198.25689 193.52347
6 198.25689 231.6275 236.53689 232.677
7 236.53689 271.4103 271.78066 270.85234
8 271.78066 308.71394 306.7157 306.7895
9 306.7157 343.81044 342.052 341.5232
10 342.052 380.9602 375.4747 376.5246
11 375.4747 418.6987 411.462 413.904
12 411.462 456.3451 447.697 450.8511
13 447.697 492.2539 483.988 486.9055
14 483.988526.8416 514.992 519.932
15 514.992 559.0936 545.95 551.146
16 545.95 590.792 576.4 581.91<
17 576.4 622.43 607.1 612.9
18 607.1 654.235 638.1 644.954
19 638.1 686.281 669.8 676.4
20 669.8 718.19 700.9 707.8
21 700.9 750.118 731.4 739.2
22 731.4 782.5 762.1 770.7
23 762.1 815.03 793.4 802.6
24 793.4 843.88 824.9 832.9
25 824.9 871.85 858.9
26 896.7 883.1
27 921.67 907.2
28 946.8
29 972.9

For a small number of alpha particles the addition of four neutrons or a p-n pair and n-n pair does not have much of an effect on binding energy. For a large number of alpha particles the case is much different.

The relevant information is the excess binding energy; the binding energy above what is accounted for by the alpha particles. This information is shown below.

The Excess Binding Energies of Nuclides Which Have a Four Nucleon Combination Added to an Alpha Nuclide
Number of
Alpha Particles
+ alpha+ 4 neutrons+ p-n &
n-n pairs
128.203836 3.112326 0.524326
235.662218 12.15049 23.07569
335.457608 18.591272 25.819272
433.025523 23.751364 26.783334
537.612031 31.191141 32.878611
638.28 33.37061 34.42011
735.24377 34.87341 34.31545
834.93504 36.93328 35.00884
935.3363 37.09474 34.8075
1033.4227 38.9082 34.4726
1135.9873 43.224 38.4293
1236.235 44.8831 39.3891
1336.291 44.5569 39.2085
1431.004 42.8536 35.944
1530.958 44.1016 36.154
1630.45 44.842 35.96
1730.7 46.03 36.5
1831.0 47.135 37.854
1931.7 48.181 38.3
2031.1 48.39 38.0
2130.5 49.218 38.3
2230.7 51.1 39.3
2331.3 52.93 40.5
2431.5 50.48 39.5

The above data are shown in the following display. The red profile is for an added alpha particle. The effect of this four nucleon combination is immediately about 35 MeV and stays at that level until an alpha nuclide of 14 is reached. The effect then falls to about 31 MeV.

The yellow profile is for the effect of four additional neutrons. This effect is small for the small alpha nuclides but it quickly rises and then increases linearly with the number of alphas. The sharp rise and then smaller rise is a shell phenomenon. The linear rise indicates that the four neutron combination is interacting with each alpha particle in a shell.

The green profile is for the effect of a combination of a p-n pair and n-n pair on binding energy. The effect is small for small alpha nuclides but rises quickly to a level of 25 MeV and then increases linearly toward a level of about 40 MeV. Again the linear increase with the number of alpha particles in a shell indicates that the p-n and n-n pair combination is interacting with each of the alpha particles in a shell.

The absence of a linear rise for the effect of an additional alpha particle indicates that the alpha particles are not all interacting with each other.

There is a hump that appears in each of the profiles indicating that it is a structural feature of the nuclides.

The most convenient way to see evidence of possible shell structures is to look at the increments in binding energy. The graphs of the increments in excess binding energies for the effect of the four neutron combination and the p-n and n-n pair combination are shown below.

These displays reveal an interaction with two to three separate shells of alpha particles. When these displays are compared with the corresponding display for the effect of an additional alpha particle it is clear that the effects of the other two four-nucleon combinations are entirely different from that of an alpha particle.

Thus the four-neutron combination and the p-n/n-n pair combination interacts with the alpha particles in a way that an additional alpha particle does not.

(To be continued.)

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