San José State University

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 The Structural Binding Energies of Alpha Nuclides and the Statistical Explanation of the Binding Energies of All Nuclides

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.

From this diagram there can be discerned critical points at #α around the levels of 4, 7 and 10, which correspond to neutron numbers of 8, 14 and 20, all magic numbers. The effects at these critical levels however influence the relationship only over short intervals so that overall the relationship appears to be linear from #α=2 to #α=14.

This suggests that the equation which would fit the data for the binding energies of the integral alpha particle nuclides is of the form

#### BE = c1#α + c2u(#α-2) + c3u(#α-14)

where u(z) is the ramp function such that if z<0 then u(z)=0 and otherwise u(z)=z. The coefficients c1 and c2 are positive and c3 negative.

A multiple regression of the binding energies of the integral alpha particle binding energies on functions of the number of alpha particles yields

#### ```BE = 29.25983#α + 5.83034u(#α-2) −4.76477u(#α-14) [22.3] [4.2] [-15.3] R² = 0.999917 ```

The figures in square brackets below the coefficients are the t-ratios for the coefficients, the ratio of the coefficient to its standard deviation. The statistical fit is quite good.

## The Effect of Additional Neutrons

Consider 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. The number of neutrons at the bendpoint of #α=5 is 10+4=14. Likewise the number of neutrons at #α=23 is 46+4=50.

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. The bump is centered at #α=12 which corresponds to 24+4=28 neutrons, a magic number.

Now consider the case of the alpha plus 24 neutron nuclides. The graph showing the increments in binding energy for each additional alpha particle is as follows.

There are sharp drops in the levels at #α=25 and #α=29. The number of neutrons for #α=29 is 58+24=82, a magic number. The number at #α=25 is 50+24=74, an anomaly, perhaps accounted for by there being a magic number, 50, of protons.

A regression of the binding energies of all 2931 nuclides on #α, u(#α-2), u(#α-14), u(#α-25) and u(#α-41) yields a coefficient of determination, R², of 0.98031.

This is fairly good but the effect of additional neutrons and protons beyond those in the alpha particles is left out. If those are included the coefficient of determination rises to 0.999435. This fit is quite remarkable.

Although the above statistical analysis was carried out in terms of the number of alpha particles the analysis 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. If neutrons were added one at a time the results would have been distorted by the effect of the creation of neutron pairs.

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 illustrated 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.

A regression based upon the number of neutrons, #n, and the number of protons, #p, with breakpoints at 4, 28, 50, 82 and 126 where the level and the slope might change yields a coefficient of determination of 0.999465. (There were no nuclide with #p≥126.) The correlation between the regression estimates and the data is then 0.999733.

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(To be continued.)