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The masses of nuclides are less than the sum of the masses of the protons and neutrons which they contain. This mass deficit translated into energy equivalence is called the binding energy of the nuclide. The binding energy of a nuclide could be composed of two components. One component could be the binding energy of substructures which are composed of protons and neutrons and another the binding energy due to the potential energy involved in putting these substructures together into an arrangement. Previous work indicates that the protons and neutrons within a nucleus form alpha particles whenever possible. Any additional protons and neutrons beyond the alpha particles substructures form pairs; neutron-neutron, neutron-proton and proton-proton. These pair formations are not mutually exclusive; i.e., a neutron may form a pair with a proton as well as with another neutron.
A major component of the binding energy of a nuclide would then be that due to the formation of alpha particles and nucleon pairs. The rest of the binding energy would be due to the configuration of those alpha particles and nucleon pairs and any excess protons or neutrons. This binding energy will be referred to as excess binding energy.
There are accepted values of binding energies for alpha particles (28.3 MeV) and for neutron-proton pairs (2.2 MeV) but no accetpted value for neutron pairs.
The excess 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. A shell is a collection of particles with the same quantum number(s) 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 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 excess binding energy by computing the increase in binding energy as the number of potential alpha particles increases.
Thus for a nuclide that could contain two to fourteen alpha particles the binding energy increases by 35.6 MeV for each additional alpha particle. This 35.6 MeV comes from the 28.3 MeV of the alpha particle itself plus about 7.3 MeV for the effect of the additonal alpha particle on the arrangement within the nucleus. It is notable that the figure of 7.3 MeV for an additional alpha particle is close to the 7.1 MeV figure for the average binding energy per nucleon in the alpha particle.
For nuclides containing fourteen to twenty five alpha partices the effect on an additional alpha particle on binding energy is an increase of 28.3 MeV for the alpha particle itself plus about 2.70 Mev for the effect of the additonal alpha particle on the arrangement within the nucleus.
The breakpoint comes at 56Ni. It has been long known that there is something special about Iron, Nickel and Cobalt nuclides. It is notable that the range over which the increment due to an additional alpha particle is about 7.3 MeV is from 2 to 14, a span of 12 alpha partices. The range for which the increment is about 2.7 MeV is from 14 to 25, a span of 11. There does not exist an integral alpha particle nuclide beyond 25. It appears that there is some arrangement of alpha particles to which an additional one may be added up to a total of twelve. This could be characterized as a shell of alpha particles. Beyond twelve there is a different shell to which alpha particles may be added.
In order to investigate this matter further some additional graphic relationships have been constructed. These, including the one above, are for
Because no accepted figure is available for the binding energy of neutron pairs the excess binding energies are computed based only upon the binding energy in excess of the alpha particles. The matter of estimating the binding energy of a neutron pair and quartet is dealt with elsewhere. The relationship of excess binding energy versus the number of alpha particles for last four of these sets of nuclides are
They all display a shell structure in which there are two alphas in the first shell, twelve in the second and a number no more than twelve in the third shell.
As with the integral alpha particle nuclides the incremental excess binding energy is not exactly constant within the shells.
There are close similarities in these patterns. To better examine the closeness of the patterns a few combinations of the patterns are shown together below.
The closeness of the patterns is striking and the differences are interesting. The closeness of the patterns in the third shells is remarkable.
Here are the first four patterns displayed together.
From the previous analysis it was not possible to establish the capacity of the third shell because it might only include be partially filled for the data available. Binding energies are only available for nuclides which are stable enough to have their masses measured. There are no such nuclides having an integral number of alpha particles beyond 25. For larger nuclides the protons are separated by a great enough distance that the nuclear force is not effective in overcoming the electrostatic repulsion. More neutrons are needed to effectively glue the protons together in the nucleus.
To investigate the capacity of the third shell the binding energies were compiled for those nuclides containing an integral number of alpha particles plus four additional neutrons. The excess binding energies were computed and the results are as shown.
The display is quite remarkable. Four shells are displayed. After the 25th alpha particle the increments are negative. This might seem surprising but it shouldn't be. At atomic numbers in the range of 90 and above (45 or more alpha particles) the binding energy must be low or negative since elements in that range are radioactive.
The values indicate that, surprisingly, the capacity of the third shell is eleven alpha particles. The incremental changes in excess binding energies displays similarities to the previous patterns but with a continuation into the fourth shell.
To obtain information on the fourth shell graphs were constructed for the binding energies of nuclides that contain an integral number of alpha particles plus eight neutrons.
The graphs of the incremental changes per additional alpha particle are
The data indicates that the capacity of the fourth shell is actually eleven alpha particles, the same as for the third shell. Eleven is a difficult number for finding some symmetrical arrangement but it should be noted that eleven alpha particles involves 22 protons and 22 neutrons. The differencce between magic numbers 28 and 50 is 22. Likewise the difference between magic numbers 82 and 126 is 44, twice 22 and four times 11.
The previous material demonstrated that the relationships between the excess binding energies of (alpha+) nuclides and the number of alpha particles are quite similar. It is now worthwhile examining those relationships in more detail. Here is the comparison of the excess binding energy for alpha nuclides and the (alpha+one neutron) nuclides.
It might have been expected that the excess binding energy for an (alpha+one neutron) nuclide would be the excess binding energy shifted upward by the binding energy accounted for by one neutron. However, the above graph, shows the effect of the additional neutron increases with the number of alpha particles. This might be explained by the extra neutron forming some sort of bond with each alpha particle. However the difference in excess binding energy is not a simple linear function of the number of alpha particles, as is shown by the graph below.
The break point in the above graph comes at 14 alpha particles. This is as it should be. The bonds of the additional neutron would be different for the alpha particles in the the different shells of alpha particles. However the effect of the additional neutron is not constant withing a shell. There is a small increase with the number of alpha particles within a shell.
The little spike comes at 20 alpha particles.
For the nuclides which alpha plus 2, 4 and 8 neutrons the relationships are similar.
The peaks in excess binding energy are reached for successively smaller number of alpha particles as the number of additional neutrons increases. This could mean that some substructure of neutrons, say a quadruplet of neutrons, fills one of the places in a shell.
(To be continued.)
The effect of additional neutrons increases with the number of alpha particles but is related to the shells to which the alpha particles belong. The effect of the additional neutrons is affected by the number of alpha particles within a shell.
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