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The Spatial Structure of the Nucleonic Shells of Nuclides

The evidence is overwhelming that the neutrons and protons of a nucleus are organized in shells. Furthermore the capacities of these shells are revealed by the so-called magic number, the numbers representing the filling of shells. The conventional magic numbers are {2, 8, 20, 28, 50, 82, 126}. There is incontroverible evidence that 6 and 14 are also magic numbers. Furthermore the existence of a simple algorithm for generating the magic numbers {2, 6, 14, 28, 50, 82, 126: suggests that 8 and 20 are in a separate category of magic numbers from the rest.

If the filled shells contain {2, 6, 14, 28, 50, 82, 126} nucleons then the capacities of the shells are given by {2, 4, 8, 14, 22, 32, 44}.

There is also evidence that the neutrons and protons in a nucleus, whenever possible, form an alpha particle of two neutrons and two protons. This makes the number of alpha particles in each shell equal to {1, 2, 4, 7, 11, 16, 22}. There is no question of what the spatial structure of the first two alpha have to be.

The Structural Binding Energies of Nuclides
as a Function of the Number of Alpha Particles

The structural binding energy of a nuclide is its binding energy less the binding energy accounted for by the formation of substructures within it. The most important substructures are the alpha particles. There can also be nucleonic spin pairs; i.e., neutron-neutron pairs, proton-proton pairs and neutron-proton (deuteron) pairs. The binding energies Of these three types of spin pairs only the are not known with any certitude. Even that of the deuteron is not measured. It is assumed to equal in value to the energy of the gamma photon emitted upon its formation. There is reason to believe that this assumption is not correct.

The Structural Binding Energies of the Alpha Nuclides

Consider all of the nuclides that could be made up entirely of alpha particles. These will be called the alpha nuclides. They are listed in the following table along with their binding energies. The binding energies of the alphas in a nuclide are just the binding energy of an alpha (28.295674 MeV) times the number of alphas in the nuclide.

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

The data on the structural binding energies are plotted below.

The bent line pattern is an indication of a shell structure. When one shell is filled a higher, lower energy shell begins to be filled. Although the incremental binding energy appears to be constant from three alphas up to 14 one closer examination, as shown below, reveals the variations.

The Binding Energies of Nuclides Which are
Alpha Nuclides Plus a Fixed Number of Neutrons

There are no alpha nuclides having atomic numbers above Tin (50). However the set of nuclides which could contain only alpha particles plus four neutrons does go beyond atomic number 50.

Again there is a bent line pattern but now there is rounding at the bend points. The levels and transition points are more easily seen in terms of the incremental binding energies.

It is notable that the transition to a higher shell does not come at 25 alphas as is the case for the alpha nuclides. This is because the crucial variables are the number of neutrons and the number of protons. As can be seen below the transition occurs when the number of neutrons is equal to 50.

One significant aspect of the above graph is the near constancy of the incremental binding energy for the shell that includes the alpha particles beyond the 14th and less than the 23rd. This means that there must be a great deal of symmetry to the spatial arrangement because the increase in binding energy when another alpha particle is added to the shell is almost the same as when the first alpha particle is put in the shell.

It is also notable that the incremental binding energy of another alpha particle is on the order of the involved in the formation of a spin pair or two.

What Holds Alpha Particles Together?

Each alpha particle has an electrostatic positive charge and hence there is an electrostatic repulsion of alpha particles for each other. Also the strong force charge of an alpha particle would result in a repulsion, at least from a distance. However an alpha particle has a proton side and a neutron side. In close proximity the proton side of one alpha particle is more attracted to the neutron side of another alpha particle that it is repelled by the proton side of the other. This is depicted schematically in the diagram below in which red represents protons and black represents neutrons.


Let s be the distance between the centers of two alpha particles oriented as in the above diagram. Let r be the distance between the midpoint of the protons and the center of the alpha particle it is part of. Likewise the distance between the midpoint of the neutron and the center of its alpha particle is r. The distances between the protons of one alpha particle and the neutrons of the other are (s+2r) and (s−2r). The distance between the protons of one alpha particle and the protons of the other is s and likewise this is the distance between the neutrons of one alpha particle and those of the other.

The force between one nucleonic cluster having a strong force charge of z1 and another of strong force charge z2 is

F = Hz1z2exp(−p/σ0)/p²

where H and σ0 are parameters and p is the distance between the centers of the clusters.

If the strong force charge of a proton is defined to be +1.0 then the strong force charge of a neutron is −2/3. This means that the value of z1z2 for the interaction between the protons of one particle and the neutrons of another particle is (2)(−4/3)=−8/3. Because of the sign of this product the force is an attraction. The two distances involved are (s+2r) and (s−2r).

The value of z1z2 for the interaction of the protons in two different particles is 4 and between the neutrons it is +4/3. Thus both of these forces are repulsions.

The net strong force between two alpha particles is then

Fs = −(8/3)H[exp(−(s+2r)/σ0)/(s+2r)² + exp(−(s-2r)/σ0)/(s-2r)²] + (4+4/3)H[exp(−s/σ0)/s²]

There is also the electrostatic repulsion between the protons of the two particles which is equal to


From all of these terms one can factor out the term (8/3)(H/(σ²s²). What is left is

[exp(−(1+2ε)ζ)/(1+2ε)² + exp(−(1-2ε)ζ)/(1-2ε)² + 2exp(−ζ)] + (1/6)(K/H)

where ζ=s/σ and ε=r/s.

Obviously this function has a singularity at ε=0.5 and that the terms in the square brackets approaches zero as ε→0, either as s→∞ or as r→0. Thus two alpha particles can be tightly bound if they are close enough together with the protons of one in close proximity to the neutrons of the other. This suggest that particles in a nucleus would have to be arranged in chains. Such chains could be closed, forming loops.

Consider again the number of alpha particles in each shell; i.e., {1, 2, 4, 7, 11, 16, 22}. The spatial arrangement could be such that each new shell is added on to the arrangement of the previous shells. Alternatively the creation of the second shell could involve the creation of an arrangement of 3=1+2 and the third an arrangement of 7=3+4. Then the fourth shell might involve the attachment of each of the new particles to a particle in the previous shell. This would require 7 for the fourth shell. For the fifth shell it is notable that its capacity of 11 is equal to 7 plus 4. This relation does not hold exactly for the sixth shell in which its capacity of 16 is slightly less than the sum of the capacities of the previous two shells, 18=11+7. For the seventh shell the capacity of 22 is far less than the sum of the capacities of the previous two shells, 27=16+11.

(To be continued.)

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