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The Formation of Alpha Particles and
Other Substructures in Nuclei

This material examines the evidence that where possible alpha particles are formed within nuclei. This is done by looking at the effect on binding energies of adding additional nucleons. When particles are added they may increase the binding energy in two ways. They may add to the binding energy through the formation of a substructure, such as a neutron-proton pair or an alpha particle, and they may add to the binding energy through the interactions of those substructures with the other substructures already in the nucleus.

Consider the binding energies of a sequence of nuclides in which first a proton is added, then a neutron, then another proton and finally a neutron. The result is very little of an increase in binding energy for the first proton, more of increase when then a neutron is added because that makes possible a proton-neutron pair. There is a further increase when a second proton is added. However the big increase comes when a second neutron is added which makes the formation of an alpha particle possible. A graph of the results is shown below.

The more curved the line the more it illustrates the effect of the formation of an alpha particle. In the graph the line is noticably less curved for the larger nuclides. This is illustrated more strikingly in the following graph.

The higher peaks are for the formation of alpha particles. The lesser peaks are for the formation of a neutron-proton pair. For the smaller nuclides the lesser peak is small to nonexistent. For the larger nuclides the lesser peaks increase in value until they are nearly as high as the major peaks. This might lead to a suspicion that for the larger nuclides only pairs are being formed. But such is not the case.

Let the nuclides that could contain an integral number of alpha particles be called the alpha nuclides. Now consider the difference in binding energies of nuclides which could contain an integral number of alpha particles plus a neutron-proton pair. The graph of the data is as follows.

The binding energy of the neutron-proton pair itself is known (or, at least thought to be known) to be 2.224573 MeV. The neutron-proton pair interacts with the alpha particles in the first shell rising quickly to 12.5 MeV for the first three alpha particles in that shell. When a fourth alpha particle is added the shell is filled and the effect of the neutron-proton pair drops back to 9.8 MeV, perhaps because there is no longer a convenient place for the neutron-proton pair as there was before the first shell was filled. As the second shell begins to fill the effect on binding energy of the neutron-proton pair is at 13.5 MeV. Most of this effect is due to the interaction of the neutron-proton pair with the elements of the filled shell. The effect rises slightly with each additional alpha particle until just before the second shell is filled. The second shell is filled with six alpha particles, which means 12 neutrons. When the number of neutrons is 20 the second shell is filled and the effect of the neutron-proton pair on binding energy drops from 14 MeV to 12.6 MeV. Again probably because the filled shell precludes the same placement of the neutron-proton pair as when that shell was not completely filled. The third shell involves 22 to 28 neutrons. As the third shell begins to fill the effect of the neutron-proton pair jumps back up to 14.9 MeV and increases with twelfth and thirteenth alpha particle but drops to 13.1 Mev when the shell is filled. The number of neutrons at that point is 28. The effect of the neutron-proton pair stays at the 13 MeV level thereafter except for a small upward blip at 38 neutrons (19 alpha particles).

The effect on binding energy of adding a second neutron-proton pair and thus making possible the creation of an alpha particle is shown below. This is the binding energy of the nuclides with two more neutron-proton pairs less the binding energy of the nuclides with one neutron-proton pair.

The sharp drops in the effects come at six neutrons, 14 neutrons and 28 neutrons. These are magic numbers. Beyond 28 neutrons the effect is approximately constant and at the 18 MeV level.

The difference of the two relations can be considered the effect on binding energy of the formation of an alpha particle from two neutron-proton pairs. The graph of this difference is as follows.

There is an interesting contrast between the effect of additional neutron-proton pairs and that of an additional neutron pair.

Whereas the effect of a neutron-proton on binding energy as a function of the number of alpha particles in the nuclide is flat, the effect of a neutron pair increases roughly linearly with the number of alpha particles in the nuclide. This is as it should be. There is an interaction of the neutron pair with each alpha particle, so the more alpha particles there are the greater the interaction and hence the greater the effect on binding energy. This also occurs for a single neutron.

This effect however appears to have a definite shell structure.

Thus the neutron-proton pairs appear to have relatively little interaction with the other substructures of a nucleus. It is not zero but is small compared with that of neutron pairs. The following graphs shows the excess binding energies in alpha nuclides; i.e., the binding energy in excess of what is accounted for by the formation of alpha particles.

Conclusions

Beyond the irregularities of filling of the first shell each neutron-proton increase the binding energy by about 12 MeV, relatively independent of the number of nucleons in the nuclide. The combination of two neutron-proton pairs into an alpha particle further increases the binding energy by about 18 MeV, 14 MeV of which is for the neutron-proton pair and 4 MeV for the formation of an alpha particle. The effect beyond 28 neutrons (14 alpha particles), is also independent of the number of nucleons in the nuclide.

The effect of a neutron or a neutron pair increases with the number of nucleons in the nuclide. Thus for a larger nuclide the effect of adding a single neutron is to increase binding energy by about 10 MeV and for adding a neutron pair the effect is about 28 MeV, about the same as adding an alpha particle. This means that the effect of the formation of a neutron pair on binding energy is about 8 MeV.

The relatively weak interaction of neutron-proton pairs with other such pairs is undoubtably due to their separation distances being such that the strong force between nucleons is nearly counterbalanced by the electrostatic repulsion between the protons.


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