San José State University |
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on the Binding Energies of Nuclides |
When a proton is added to a nuclide there are two effects on the potential energy and consequently on the binding energy. The largest effect is due to the nuclear strong force attraction of the proton for the other nucleons. The other effect is due to the repulsion of the positive charge of the proton for the other protons of the nuclide. The effect of the repulsion is to reduce the binding energy.
Consider the nuclides that could contain an integral number of alpha particles. Hereafter these will be called alpha nuclides. Then The effects of adding a proton or a neutron on binding energy are irregular, but their difference is a rather smooth curve.
What is plotted in the graph as the little red squares is the binding energy of an alpha + one neutron nuclide minus the the binding energy of an alpha + one proton nuclide. This is the decrease in binding energy due to the presence of a positive charge alone, or at least a plausible first approximation to that quantity.
The effect of two positive charges can be computed as (α+2neutrons) − (α+2protons). This is shown below with the effect of one positive charge and the effect of one positive charge doubled.
As can be seen from the graph the effect of two positive charges is close to being equal to twice the effect of one positive charge. There is a slight shortfall. This shortfall would be for the interaction of the two positive charges themselves, as opposed to the interaction with the rest of the nucleons.
When the effects of three and four positive charges are computed and compared with triple and quadruple the effect of one positive charge it found that there is a small shortfall.
The shortfall for four positive charges is noticably larger than for three. This is as it should be because there are more interaction pairs for four charges than for three or two charges. For two charges there is just one pair. (These interaction pairs are often called bonds.) For three charges there are three bonds and for four there are six. However for four particles at the corner of a square the distance for the diagonal bonds is √2 times the distance along the edges. For the electrostatic force the bonds would have to be weighted by the inverse square of the distances. This means that for four positive charges the effective number of bonds is (4+½+½)=5.
Below is the graph of the binding energy shortfalls for four, three and two positive charges.
The data for four positive charges only goes up to 14 alpha particles. That for three goes to 17 alpha particles.
The interesting quantities are the ratios of the effect for four positive charges and the effect for three positive charges. Here are the results of the computation.
The simple counting of the number of bonds suggested that the ratio of the effect for three positive charges to the effect for two indicates that the ratio should be 3 and that for four charges it should be five. The simple theory gives values that are the right order of magnitude.
If the ratio of the effect for four positive charges to the effect for three is computed the values are very close to the ratio of the effect for three to the effect for two. This is shown in the graph below, which is the same graph as the previous one except for the ratio for 4+ to 3+ shown as the upper edge of the blue area.
Thus, the effect for four is roughly the square of the effect for three.
Since the positive charges are only affected by the inverse distance squared force it is easy to convert potential energy into distances. The formula for the relation between the potential in Mev and the distance in fermi is
The average distances between the positive charges are then given in the following graph.
When the analysis is extending to five and six positive charges it is seen in the graph below that there are drastic changes in the phenomena. The effects for five and six positive charges are much bigger and the profile as a function of the number of alpha particles entirely different than the effects for three and four.
(To be continued.)
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