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The Energies Involved in the Formation of Nucleon Pairs

Neutron pairs, proton pairs and perhaps neutron-proton pairs are formed within nuclei. There are a number of questions to be answered concerning this various nucleon pairs. Among them are:

The last question can be dealt with very easily. Below is a display of the incremental binding energies of the isotopes of Calcium.

As can be seen the amplitude of the fluctuations is approximately constant within a shell but decreases as additional neutrons go into a new shell. These fluctuations in incremental binding energy are somehow related to the energy in the formation of pairs, in this case neutron pairs. This could be a direct relation involving the potential energy lost in the formation of the pair. On the other hand, it might be that the binding energy relates only to changes in the potential energies involved in the proximity of the nucleons. In this case the change in the binding energy arising from the formation of a pair would come from the adjustments in separation distances of the nucleons which the pair formation makes possible. The change in potential energy involved in a pair formation would seem to be independent of all conditions of the nucleons, but as shown above for the case of the Calcium isotopes, that is not true. Thus the enhancement of binding energy arising from the formation of a nucleon pair would seem to be the result of the adjustment of separation distances. In outer shells where the separation distances of the nucleons are greater there would be less of an effect than for inner shells.

The energy involved in the formation of a neutron pair could be approximated by taking the increase in incremental binding energy when another neutron is added to a nuclide with an odd number of neutrons. However there is a general decline in incremental binding energy with the number of neutrons so the odd-even difference combines the energy of pair formation with the decline in incremental binding energy. Likewise the even-odd difference would conflate the two effects. The obvious thing to do is to average the adjacent low values of incremental binding energy and subtract that value from the high value. Likewise the adjacent high values can be averaged and the low value subtracted from that average. When this procedure is carried out for Calcium the graph of the results is as shown below.

The peaks centered around N=20 and N=28 should be ignored because they reflect the transition to higher shells and not pair formation. As can be seen the energy involved in the formation of a neutron pair in the lower shell (part of the 15 to 20 neutron shell) is about 3 Mev. In the middle shell (the 21 to 28 neutron shell) it is 3.5 to 3.0 Mev. And in the higher shell (part of the 29 to 50 shell) it is about 1 MeV.

The corresponding display for the nuclides with 20 neutrons and varying numbers of protons (the isotones) is shown below.

This case spans a different set of shells than the P=20 (Calcium) case, but there is the same near constancy within shells. There appears to be a shell transition at P=10 rather than P=8.

The estimates of the energies involved in the formation of proton pairs are shown below.

As indicated previously the levels of the graph centered around the points of transition between shells at P=14 and P=20 should be ignored. In this case this also seems to apply for P=10. The energy involved in the formation of a proton pair in the first shell is about 5 Mev. In the 15 to 20 proton shell the energy for a proton pair formation is about 3 MeV and likewise for the 21 to 28 proton shell. Thus the values of pair formation for the 21 to 28 nucleon shells are roughly the same for proton and neutron pair formations.

To get a good correspondence of the shells with the display for Calcium (P=20) it is necessary to go to N=28.

The estimates of the energy for the formation of a proton pair are given below.

The values for the formation of neutron pairs and proton pairs are roughly the same for the 21 to 28 nucleon shells. For the lower and the higher shells the values for the formation of a proton pair is significantly higher than for the formation of a neutron pair. For the 29 to 50 shell the value for proton pair formation is about 2.5 MeV whereas for the formation of a neutron pair in that shell the value is only about 1 MeV.

(To be continued.)


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