San José State University

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 The Structural Binding Energies of Nuclides and the Effects of Additional Neutrons

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.

## 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. With the lesson of this case we may go back to the alpha plus 1 neutron nuclides. The bend points are at 5 alpha particles and 24 alpha particles.

In the graph below the differences in binding energies between the alpha plus 1 neutron nuclides and the alpha nuclides are plotted along with the regression estimates of those differences against the number of alpha particles. The fit is not as good as for the alpha plus 4 neutrons but clearly the regression estimates show the structure of the relationship.

Now the effect of neutron pairs can be examined. The excess binding energy due to two neutrons, one neutron is as shown below. Even with only one neutron pair the downturn for the fourth shell of alpha particles is in evidence.

The structural binding energy for one neutron pair is This is the same type of relation that was seen for the alpha plus 1 neutron nuclides and the alpha plus 4 neutron nuclides.

The notion of the structural binding energy of a particle can be extended. The effect of adding a second neutron pair is the difference between the binding energy of an alpha plus 4 neutron nuclide and an alpha plus 2 neutron nuclide. Some of the increase in binding energy is due to the formation of the neutron pair. The value of the neutron pair formation is not known. In contrast the value for a proton-neutron pair formation is known to 2.22457 MeV.

The results of this computation of the effects of a second additional neutron pair are shown below. This appears to be a different type of relation than was seen above. Here when a critical level of the number of alpha particles is reached there is jump upward or downward as well as a change in slope. The magnitudes of the jumps can be easily estimated using so-called dummy variables. A dummy variable is a {0, 1} variable whose value is 0 below a critical level and 1 at or above it.

A regression equation was estimated for the data on the effect of a second neutron pair on binding energy using 4 and 24 as the critical levels. The data corresponding to the bump on the linear relationship were left out of the regression. The estimates for the binding energy due to the second neutron pair are shown for comparison to the actual values are shown below. The results of the regression analysis indicate for the first three alpha particles the binding energy due to the second neutron pair increases 1.66 MeV for each additional alpha particle, then there is a jump of 6.21 MeV. Thereafter the binding energy increases by 0.67 MeV for each additional alpha particle until there are 24 alpha particles. Thereafter there is drop of 3.90 MeV and an increase of 0.51 MeV for each additional alpha particle. The coefficient of determination (R²) for the regression is 0.995373.

The difference between the binding energies of the alpha plus 6 neutron nuclides and the alpha plus 4 neutron nuclides has the same general form as the one for the 2 neutrons and so for the case of more additional neutron pairs, as is seen in the displays below. The red squares are for the first neutron pair added to an alpha nuclide. The upper edge of the yellow area represents the effect of the second neutron pair. The effect of the third neutron pair is shown as the upper edge of the green area and the fourth by the upper edge of the blue area. The slope of the profile increases with the number of additional neutron pairs. In this display the red squares are the effect of the fifth added neutron pair. The upper edge of the yellow area is for the effect of the sixth neutron pair, the upper edge of the green area for the seventh, the blue area for the eighth and the violet area for the ninth. The slope of the profile doesn't change much for additional neutron pairs.

Generally the effect of an additional neutron pair is to shift the profile to the left by one unit and downward. There is also a sharpening of the peaks in the profile.

The data for the effects of the addition of the first through fourth neutron pairs were combined and a regression equation estimated based upon critical points at 5 and 25 alpha particles with the critical points shifted down for each neutron pair. The regression estimates are plotted along with the data in the following graphs.    The coefficient of determination (R²) for the combined regression is 0.9306. The results for the fourth neutron pair indicates that there may be another critical point at 32 alpha particles.

The pattern is continued for the addition of the tenth through the twelfth neutron pair, as shown below. ## Conclusions

The profile of impacts of additional neutron pairs as a function of the number of alpha particles in the nuclide exhibits a shell structure.

• Each additional neutron pair raises the binding energy by a smaller amount than the previous one.
• The critical point for the profile is shifted backward one alpha particle for each additional neutron pair.
• The slope of the relation between additional binding energy and the number of alpha particles becomes steeper for each additional neutron pair, at least for the first few 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.)