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The Cross Difference of the Excess Binding Energies
Is Equal to the Interaction Energy of the Last Alpha
Particle with the Last Neutron 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. There is a lesser drop after 20 alpha particles which corresponds to 40 protons and neutrons, a number not generally designated as a magic number. It happens that 8 and 20 are the sum of the two previous numbers in the sequence {2, 6, 14, 28}. The number 40 is approximately the sum of the two magic numbers 14 and 28.

The same sort of computations can be made for the nuclides that could contain an integral number of alpha particles plus one neutron. The graphs of the results arel shown below.

In this case the number of neutrons is one more than the number of protons and the correspondence of the number of alpha particles with the neutron and proton shells is different. Also the increase in the number of neutrons changes the values of the IXSBE for the alpha particles. The difference is called the cross difference. It corresponds to the cross derivative in calculus. If the XSBE is a functions of the number of alpha particles a and the number of extra neutrons n, then the cross derivative is ∂²XSBE/∂n∂a. The graph of the cross differences is shown below.

These cross differences have some remarkable properties. For example, the same values would arise if first the incremental differences were calculated for an increase in the number of extra neutrons and the differences in these incremental values were computed for an increase in the number of alpha particles. A much more important property is considered below.

The increase with respect to the number of extra neutrons in the incremental excess binding energies of the alpha particles is equal to the interaction energy of the last alpha particle with the last extra neutron.

Proof:
Consider a nuclide with a alpha particles and n extra neutrons. The binding energy of that nuclide represents the net sum of the interactions of all a alpha particles with each other, all n extra neutrons with each other and all na interactions of the a alpha particles with the n extra neutrons.

   

The black squares indicate there are not any interactions of a particle with itself.

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 extra neutrons and that of the nuclide with a-1 alpha particles and n extra neutrons. In the diagrams below the interactions of the nuclide with (a-1) alpha particles and n extra neutrons are shown in color.

   

The subtraction eliminates all the interactions of the n extra neutrons with each other. It also eliminates the interactions of the (a-1) alpha particles with each other and the (a-1) alpha particles with the n extra neutrons. What is left is the interaction of the a-th alpha particle with the other (a-1) alpha particles and the interaction of the a-th alpha particle with the n-th extra neutron.

  

Now consider the difference of the IXSBE for a alpha particles and n extra neutrons and the IXSBE for a alpha particles and (n-1) extra neutrons. In the diagrams below the interactions for the IXSBE for the nuclide with (n-1) extra neutrons are shown colored.

  

The subtraction eliminates the interactions of the a-th alpha particle with the other (a-1) alpha particles. It also eliminates the interactions of the a-th alpha particle with the (n-1) extra neutrons. What is left is the interaction of the a-th alpha particle with the n-th extra neutron.

Conclusions

Consider again the cross differences in the excess binding energies which were shown above.

The most important property of these cross differences is that they are positive (except for the initial point). This means that the force between alpha particles and neutrons is an attraction. There is also a shell structure which indicates that the interaction energy between alpha particles and extra neutrons is roughly constant within shells. From the fourth alpha particle to the fifteen the interaction energy is roughly 8 MeV. Above the fifteenth it is about 3 MeV.

It is no surprise that there is an attractive force between a neutron and an alpha particle. The conventional theory holds that a neutron would be attracted to all four nucleons in an alpha particle. However previous studies concluded that neutrons are repelled by other neutrons but attracted to protons. If the nucleonic charge of a neutron is equal in magnitude to that of a proton then an alpha particle there would be no net force between a neutron and an alpha particle. There being a positive force between the two indicates that the nucleonic charge of a neutron is smaller in magnitude as well as of opposite sign. Previous studies indicate that if the nucleonic charge of the proton is taken to be +1 then the nucleonic charge of the neutrons is −3/4. There are more studies that conclude that the nucleonic charge of the neutrons is −2/3.


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