San José State University

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 The Structural Binding Energies of Nuclides and the Effects of Additional Neutrons(Version 2)

The conventional binding energy of a nuclide is its mass deficit represented in energy terms. Its mass deficit is the difference between the sum of the masses of its constituent protons and neutrons and its mass. This approach leaves out the fact that some of the mass deficit is due to the mass deficits of substructures and some is due to the arrangement of the substructures in the nuclide. It is this latter component that is of primary interest concerning the structure of nuclides. It can be computed by deducting from the binding energy of a nuclide the combined binding energies of its substructures.

Nuclides undoubtably contain nucleon pairs. There is also strong, perhaps overwhelming, evidence that nuclides contain alpha particles.

The binding energy of a nuclide in excess of the binding energies of its substructures will be called its structural binding energy. The place to start the computation of the structural binding energies is with the nuclides which could contain an integral number of alpha particles. Hereafter these nuclides will be called the alpha nuclides. Let #α be the number of alpha particles which could be contained in a nuclide. The excess or structural binding energy of an alpha nuclides is its binding energy minus (28.29567)*#α, where 28.29567 MeV is the binding energy of an alpha particle. The results are plotted in the graph below. This displays a definite shell structure. There are bendpoints at #α=2 and at #α=14. This means that first shell can contain two alpha particles and the second can contain 12. The third shell can contain at least 11 alpha particles.

Note that two alpha particles means the number of neutrons, #n, is 4. Likewise at the second bendpoint the number of neutrons is 28 and at #α=25 the number of neutrons is 50. The neutron numbers 4,28 and 50 are among the magic numbers of neutrons. In the incremental excess binding energies are computed we see the values are not constant. ## The Effect of Additional Neutrons

The next most simple structure is for the nuclides which contain an integral number of alpha particles plus one neutron. The computed excess binding energies for this set of nuclides are shown below as a function of the number of alpha particles in the nuclide. This displays the same sort of shell structure as does the graph for the alpha nuclides. This computed excess binding energy includes the structural component of the binding energies of the alpha nuclides along with that solely due to the additional neutron. The way to eliminate the structural binding energy of the alpha nuclides is to subtract from the binding energy of the alpha plus 1 neutron nuclide the binding energy of the corresponding alpha nuclide. The results of this computation are displayed below. Before going further with this case let us first move on to the case of the alpha plus 4 neutrons nuclides. The graphical displays for this case are  The first graph displays a shell structure with four shells. The second graph also displays a shell structure. In this case the bendpoints are at 5 alpha particles and 23 alpha particles. There is a bump on the line segment for 11, 12 and 13 alpha particles which shows up in other cases and is apparently connected with some structural feature of nuclei. Those data points cannot be matched using linear regression and the attempt to do so will distort the estimates of the parameters so they were left out of the regression analysis. Using the results of the regression analysis value of the differences in binding energy were computed and they are displayed with the actual data. The fit is quite remarkable.

Previus work indicates that the crucial variable is not the number of alpha particles but instead the number of neutron pairs in the nuclide after the neutron pairs are added. The graphs for the cases of adding one neutron pair through adding nine are shown below.   In the top graph the peaks correspond to nuclides having 10, 14 and 25 neutron pairs; i.e., or 20, 28 and 50 neutrons. In the middle graph the peaks correspond to nuclides having 14 and 25 pairs which means 28 and 50 neutrons. In the bottom graphs the peaks are at 25 and 41 pairs or 50 and 82 neutrons. These are nuclear magic numbers.

From the above graphs it is seen that as more neutron pairs are added the effect on binding energy appears to decrease in proportion to the number of added neutron pairs.

The effect of additional neutron pairs is seen by holding the number of alpha particles constant and varying the number of neutron pairs. This is shown below. The sawtooth pattern comes from adding one neutron at a time. The peaks correspond to neutron pairs. The level drops when the number of neutrons reaches the critical level of 50, one of the nuclear magic numbers.

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(To be continued.)