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The Second Difference of the Excess Binding Energies
with Respect to the Number of Alpha Particles Is Equal
to the Interaction Energy of the Last Alpha Particle
with the Next to Last Alpha Particle in the Nuclide

Incremental Binding Energies

The mass of a nuclide, such as helium 4, the alpha particle, is less than the masses of the two neutrons and two protons of which it is composed. The difference is called the mass deficit and that mass deficit expressed in energy units via the Einstein formula E=mc² is called the binding energy. The binding energies have been measured for almost three thousand nuclides. If the some of the protons and neutrons form alpha particles within a nucleus then the binding energy is composed of two parts. One part is the binding energy involved in the formation of the alpha particles and a second part due to the arrangement of the alpha particles and extra protons and neutrons within the nucleus. The binding energy of an alpha particle is 28.29567 million electron volts (MeV). The excess binding energy (XSBE) of a nuclide is its binding energy less 28.29567 MeV times the number of alpha particles which can be formed within the nuclide.

The incremental excess binding energy (IXSBE) of a particle in a nuclide is the difference in the excess binding energy of that nuclide and the nuclide containing one less particle of the same type. Shown below is the computations for the excess binding energies of the nuclides which could contain exactly an integral number of alpha particlles. Such nuclides will hereafter be referred to as alpha nuclides.

The Binding Energies of Nuclei Which Could
Contain an Integral Number of Alpha Particles
Element Neutrons Protons Binding
Energy
Number of
Alpha Particles
Binding
Energy of
alphas
Difference
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

Protons and neutrons are arranged separately in shells. The numbers corresponding to the shells filled to full capacity are known as the nuclear magic numbers. Conventionally the magic numbers are {2, 8, 20, 28, 50, 82, 126}, but a case can be made for the magic numbers being instead {2, 6, 14, 28, 50, 82, 126} with 8 and 20 being in a different category of magic numbers. For more on this see Magic Numbers.

The structure of the nuclear shells, both for neutrons and protons, is given in the following table.

Shell
Number
12345678
Capacity2481422324458
Range1 to 23 to 67 to 1415 to 2829 to 5051 to 8283 to 126127 to 184

If the nucleons are combined into alpha particles then the shell structure of alpha particles that is compatible with the neutron and proton shells is

Shell
Number
12345678
Capacity124711162229
Range1 2 to 3 4 to 7 8 to 1415 to 2526 to 4142 to 6364 to 92

A plot of the excess binding energies for the alpha nuclides shows a shell structure.

The incremental excess binding energy (IXSBE) shows more detail and that the range from three alpha particles to 14 is made up of more than one shell.

This graph shows sharp drops at 4, 7, 10 and 14; which correspond with 8, 14, 20 and 28 protons and neutrons. These are magic numbers.

The graph shows neatly the magicality of 14 and 20 where the IBE drops sharply. A sharp drop in the IBE is an indication that a shell or subshell has been filled and additional nucleons are going into a higher shell. The sawtooth pattern is due to the energy involved in the formation of spin pairs.

The Nature of the Incremental Binding Energy of the Protons

The increase in the incremental excess binding energy of an alpha particle as a result of an increase in the number of alpha particlesis equal to the interaction energy of the last alpha particle with the next to last last alpha particle, provided these two are in the same alpha particle shell.

Rationale:
Consider a nuclide with a alpha particles and n neutrons. The binding energy of that nuclide represents the net sum of the interaction energies of all a alpha particles with each other, all n neutrons with each other and all na interactions of a alpha particles with the n neutrons. Below is a schematic depiction of the interactions.

  

The black squares are to indicate that there is no interaction of a particle with itself.

What is given below is the interactions for a nuclide of a alpha particles and n neutrons overlaid with those of a nuclide with (a-1) alpha particles and n neutrons, shown in color.

  

The incremental excess binding energy of an alpha particle is the difference in the excess binding energy of the nuclide with a alpha particles and n neutrons and that of the nuclide with (a-1) alpha particles and n neutrons. When the subtraction is carried out the interactions of the n neutrons with each other are entirely eliminated. It also eliminates the interactions of the (a-1) alpha particles with each other and the (a-1) alpha particles with the n neutrons. The interactions which are left after the subtraction are the squares shown in white above.

Now consider the incremental binding energy for the nuclide with (a-1) alpha particles and n neutrons. The interactions for this IXSBE are shown in color along with those for the IXSBE for a neutron in a nuclide with a alpha particles and n neutrons shown in white.

 

Now consider the difference of the IXSBE for a alpha particles and n neutrons and the IXSBE for (a-1) alpha particles and n neutrons.

 

The subtraction of the IXSBE for (a-1) alpha particles and n neutrons from the IXSBE for a alpha particles and n neutrons depends upon the magnitude of the interaction of the (a-1)-th alpha particles with the different alpha particles compared to the interaction of the a-th alpha particle with those same alpha particles. Visually this is the subtraction the values in the blue squares from the white squares on the same level. When the a-th and the (a-1)-th alpha particles are in the same shell the magnitude of the interactions with any other proton are, to the first order of approximation, equal. Thus the interactions with the n neutrons are entirely eliminated. Likewise for the first (a-2) alpha particles. All that is left is the interaction of the a-th alpha particle with the (a-1)-th alpha particle.

The Second Differences of the Excess Binding Energies for the Alpha Nuclides

The results of the computation for the alpha nuclides are given in the following graph.

The spikes in the values occur at changes in shell levels. Within a shell the values are relatively constant and slightly negative.

The same computations for the nuclides that could contain an integral number of alpha particles plus one extra neutron produce essentially the same results as for the alpha nuclides.

A plot of the data for the alpha nuclide and the alpha plus 1 neutron nuclides shows how close the two are and that the irregularities are intrinsic rather than random.

Conclusion

The conclusion to be drawn is that the second differences with respect to the number of alpha particles are very small within a shell. The second differences represent the interaction of the last alpha particle with the previous one in the nuclide. The interaction of the last two alpha particles is almost zero indicating neither an attraction nor a repulsion between the alpha particles of a nucleus.

The interaction binding energy for the interaction of the last last alpha particle with the last extra neutron shows much larger values.

If the conventional theory of nucleons were correct the force and potential energy between two alpha particles would, at the same distance, be four times larger than that between a neutron and an alpha particle. If the nucleonic charge of a neutron were equal in magnitude to that of a proton but opposite in sign an alpha particle would be neutral concerning the nuclear strong force. There would be no interaction energy between an alpha particle and a neutron or other alpha particle. If the nucleonic charge of a neutron is somewhat smaller in magnitude as well as opposite in sign then an alpha particle would have slightly "positive" charge. There would be some force and potential energy involved in the interaction of a neutron and an alpha particle and lesser amount between two alpha particles.

The force and potential energy depends upon the separation distance between the particles. If the alpha particles are arranged in a sequence involving a maximum separation of successive particles then the interaction binding energies would be relative small. In small nuclides there would be littcxle way for the alpha particles to put much distance between themselves.


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