Second Differences

There is a theorem which says that the interaction energy of the last particle of a particular type to be added to a nucleus with the next-to-last particle of that type is closely related to the second difference of the interaction energy of nuclides with respect to the number of particles of that type. Specifically it says that the second difference is an upper bound to the interaction energy of the last two particles to be added to the nucleus. Since the second difference of binding energies is almost always negative the theorem means that the binding energy of the interaction of the last two particles of a particular type is at least as negative as the second difference. This is important because the negativity of the interaction of two particles is evidence that the force between them is a repulsion.

The analysis is limited to nuclides that consist entirely of proton-proton pairs and neutron-neutron pairs. This is to eliminate the distracting influence of the formation of those two types of spin pairs. However the formation of proton-neutron spin pairs still affect the results. Here are the second differences arranged as much as possible by nucleonic shells.

Let p and n denote the numbers of proton spin pairs and neutron spin pairs, respectively. The large magnitude of the second difference for n=2 and p=3 is due to what may be called the p=n effect. Wh en p<n the addition of another proton pair results in the formation of two proton-neutron pairs, and more importantly, an alpha module. An alpha module is a chain in which a proton is linked to a neutron which is linked to another neutron which is linked to a proton. The simplest alpha module chain is an alpha particle. A deuteron, a proton-neutron spin pair, has a binding energy of 2.2 MeV. When two deuterons are combined into an alpha particle the binding energy jumps to 28.3 MeV. When n>p the addition of another proton spin pair does not result in the formation of an alpha module and the incremental binding energy drops accordingly.

Note that all of the values are negative.

Here is a sample illustration from the above data.

The negative values are reflective of the repulsion involved in the interaction of two proton pairs. There are apparently two plateaus in the shell containing the 26th through 41st proton spin pairs. Those levels are roughly −2 MeV and −1.6 MeV.

In study relating to the interaction of neutron pairs a case was found in which the interaction of neutron spin pairs was found to be approximately 0.9 MeV over a wide range of the number of neutron spin pairs. If the nucleonic charge of a proton is taken as 1 and that of a neutron as q then the nucleonic charges of a proton spin and a neutron spin pair are 2 and 2q, respectively. The binding energy due to the interaction of two neutron spin pairs should be proportional to 4q², whereas that for two proton pairs should be proportional to 4. If the distances involved in the interactions are the same the ratio of the value for neutron pairs to that for proton pairs should be equal to q². The ratio of 0.9 to 2.0 gives a value of q of −0.6705 whereas the ratio of 0.9 to 1.6 gives a value of q of −0.75, or hence values of −2/3 and −3/4.

(To be continued.)