Consider all the nuclides which could contain an integral number of alpha particles. (Hereafter these are called alpha nuclides.) The binding energies of these alpha nuclides were compiled. Then the binding energies of all nuclides which could contain an integral number of alpha particles plus a neutron-proton pair were compiled. The differences of these series are the effects of a neutron-proton pair on the binding energies of nuclides. To get the effect excluding the binding energy of the pair itself the value of 2.224573 MeV is deducted from each value. The resulting values, called the structural binding, are shown below.

The data on the structural binding energy due to the neutron-proton pair are plotted in the following graph.

The effect rises because of the interaction of the neutron-proton pair with the other particles in the nuclei. The value reaches a peak for the number of neutrons (#n) equal to 6. It drops back for #n equal to 8 and then rises again. There is a slight upward shift for #n=14 and then a drop at #n=20 with a rise back up again. There is a drop again at #n=28. These points of rise or fall are all magic numbers. Beyond #n=28 there is slight drop at #n=30 and a rise and fall for #n=32 and then a rise for #n=38 and a fall for #n=40. This is a pattern that prevails for the other nucleon pairs as well.

The general level of the effect for #n=10 and beyond is about 11 MeV. If the binding energy of two neutron-proton pairs is deducted from the binding energy of the alpha particle the result is about 28.29-2(2.22)=23.85 MeV. Half of this is 11.93 MeV. Thus the effect on binding energy of a neutron-proton pair is roughly equal to half of an alpha particle.

One approach to explaining the effect of an additional neutron-proton pair is to compute the effects of adding separately a proton and a neutron and then comparing the sum of these effects with the effect of adding a neutron-proton pair. This comparison is shown in the graph below.

The effect of adding a single neutron is the upper edge of the yellow area. The upper edge of the green area is the effect of adding a single proton. Their sum is shown at the line of red squares. The sum is remarkably regular, being nearly linear. A quadratic regression of the sum on the number of neutrons has a coefficient of determination (R²) of 0.99745 with 21 degrees of freedom.

A comparison of the sum of the separate effects of adding a neutron and proton with the effect of adding a neutron-proton pair shows that the two are not closely related.

The difference of the two relations is of some interest. It shows the same dependence on critical points at magic numbers as does the effect of a neutron-proton pair.

The slope of the relation for #n beyond 6 is remarkably constant. A regression having bendpoints at #n=2 and #n=6, with drops at #n=8 and #n=20 and rises for #n over the interval from #n=22 to #n=26 and for #n=38. This regression has a coefficient of determination (R²) of 0.9933 with 15 degrees of freedom. The standard error of the estimate is 0.2989 MeV. The regression estimates are shown below along with the data.

The regression provides the empirical value of the binding energy of the neutron-proton pair because there is a data point with all of the explanatory variables equal to zero. Otherwise there is nothing in the analysis that would indicate that the binding energy of a neutron-proton pair is 2.224573 MeV.

(To be continued.)

The Effect of a Neutron Pair
on the Binding Energy of a Nuclide

The first step in investigating the effect on binding energy of a neutron pair is to compile the binding energies of all nuclides which could contain an integral number of alpha particles plus two neutrons. From this the binding energies of the alpha nuclides are deducted. The results are shown below.

This differs from the case for the neutron-proton pair in that there is rise in the effect with the number of alpha particles in the nuclide where for the neutron-proton pair the relation is pretty much flat. This means that the neutron pair interacts with each alpha particle whereas for the neutron-proton pair the net interaction is zero. This may be because for the neutron-proton pair the effect of the nuclear force is cancelled out by the electrostatic repulsion.

The more detailed analysis of the effect of a neutron pair is obtained by computing the effect of adding one neutron to the alpha nuclides and then the effect of adding a second one. The difference between the effect of adding a second neutron and the effect of adding the first should be related to the binding energy of the neutron pair. This is shown below.

The red squares represent the effect on binding energy of the second neutron. The effect of adding the first neutron is the upper edge of the yellow area. The difference is the upper edge of the green area. The level of this difference is not constant but seems to be tending asymptotically toward a constant level. On the upper range it is fluctuating between 2.3 and 2.6 MeV. The average of these two levels would seem to be the most reasonable estimate of the binding energy of the neutron pair.

An alternate approach to estimating the binding energy of a neutron pair yields a value of 2.569765 MeV. This is based upon the effect binding energy of switching one proton to a neutron. For the details see Binding Energy of Mirror Nuclices.

(To be continued.)

A Proton-Pair

A proton pair is the Helium-2 isotope (2 protons, 0 neutrons) which decays by disintegration into two protons with a half-life of 3×10^-27 seconds. It is not stable enough to have its mass measured and thus its binding energy determined.

When one proton is added to the alpha nuclides generally the binding energy decreases. In the graph below the effect of the first neutron is shown as the lower edge of the yellow area. The effect of the second neutron is shown as the red squares and the difference between the effect of the second neutron and the first is shown as the upper edge of the green area. The difference does not seem to be asymptotically approaching any positive value. The binding energy of a proton pair is apparently less than 1 MeV and possibly is zero.

The notably aspect of the difference is that it displays the same pattern as the differences for the other two nucleon pairs; i.e., bendpoints at #n=2 and #n=6 with point drops for #n=8 and #n=20, an upward shift at #n=10, an upward shift for #n=22, 24 and 26 and a downward shift at #n=28.

A regression was carried based up the above pattern. What is notable is that this regression requires the slope to be constant for #n above 10. The results are shown below, where the red squares are the data and the upper edge of the yellow area gives the estimates from the regression. The cofficient of determination (R²) for the regression is 0.9889 with 11 degrees of freedom.

(To be continued.)