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the Incremental Structural Binding
Energies of Alpha Particles in Nuclides
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 structural binding energy (SBE).
There are accepted values of binding energies for alpha particles (28.3 MeV) and for neutron-proton pairs (2.2 MeV) but no accepted value for neutron pairs.
The structural binding energy for those nuclides that could contain an integral number of alpha particles has an interesting form.
The binding energy of a nuclide which could contain multiple alpha particles is in excess of the binding energies of the alpha particles it might contain. The above graph suggests that there are shell structures of the alpha particles within the nuclei. 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 structural 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.
The numerical stability of the increments in structural binding energy can be examined by computing the increase in binding energy as the number of potential alpha particles increases. This is the incremental structural binding energy (ISBE) of an alpha particle.
The further details of this line of analysis are covered in the Alpha Particle Structure of Nuclides. Here the focus is on how the number of neutrons in excess of those contained in the alpha particles affect the incremental structural binding energies of an alpha particle.
The graph of the incremental structural binding energies of alpha particles is the key to the spatial arrangement of nuclei. To see that the fluctuations are not just random consider the the same sort of graph for nuclides which a made up of an integral number of alpha particles plus one neutron.
Here is a graph with the previous two graphs superimposed.
The pattern is essentially the same for the two cases. There is clearly a different shell for more than 14 alpha particles. The sharp drop in ISBE occurs after 14 alpha particles for both cases. For less than 14 alpha particles the pattern is more irregular but the relative maxima and relative minima occur at the same or nearly the same values of the numbers of alpha particles.
Because no accepted figure is available for the binding energy of neutron pairs the structural binding energies are computed based only upon the binding energy in excess of that involved in the formation of the alpha particles.
The following graphs show the dependence of the incremental structural binding energy (ISBE) of an alpha particle on the number of excess neutrons in the nuclide.
For the case of nuclides with two alpha particles there is an abrupt increase in ISBE when a second neutron is added, possibly due to the effect of the formation of a neutron spin pair. However there is not the same level of increase when a fourth neutron is added.
In the third graph the case of 14 alpha particles reflects the transition between shells. For the cases beyond 14 alpha particles the values of ISBE are all lower than for the shell of three to fourteen alpha particles and roughly the same.
These graphs are based upon the data in the following table.
Incremental Structural Binding Energies of Alpha Particles Number of Excess Neutrons 0 1 2 3 4 #α 2 -0.091838 2.459226 7.412426 8.365326 8.946326 3 7.366544 10.647489 12.011634 12.725926 13.807326 4 7.161934 6.358923 6.226818 8.964726 12.322026 5 4.729849 7.347636 9.667236 10.911856 12.169626 6 9.316357 9.885916 10.615046 11.857776 11.495826 7 9.984326 11.127206 10.643326 10.787376 11.487126 8 6.948096 7.116106 7.923856 8.322296 9.007966 9 6.639366 6.786706 7.207866 6.820326 6.800826 10 7.040626 6.614426 6.256926 7.591626 8.854086 11 5.127026 6.294326 8.003426 8.948126 9.442826 12 7.691626 8.743626 8.554026 8.938026 9.350726 13 7.939326 8.040526 8.418926 8.455126 7.613126 14 7.995326 7.559126 6.399426 6.101026 6.292026 15 2.708326 2.692326 3.369526 3.483026 3.956326 16 2.662326 2.491326 2.875326 2.869326 3.402726 17 2.154326 2.314326 2.744326 2.867326 3.342326 18 2.404326 3.014326 2.654326 3.570326 3.509326 19 2.704326 3.514326 3.432326 3.661326 3.750326 20 3.404326 3.264326 3.396326 3.411326 3.613326 21 2.804326 2.704326 2.804326 3.654326 3.632326 22 2.204326 2.804326 3.304326 3.264326 4.086326 23 2.404326 2.604326 3.404326 4.004326 4.234326 24 3.004326 3.404326 3.504326 2.164326 0.554326 25 3.204326 1.304326 -0.295674 -0.455674 -0.325674 26 -4.295674 -3.995674 -3.445674 27 -3.895674 -3.795674 -3.325674 28 -3.595674 -3.195674 -3.165674 29 -2.295674 -2.195674
(To be continued.)
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