San José State University
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The Pattern of the Impact
of Additional Neutrons on the
Binding Energies of Nuclides

The binding energies of nuclides can be compared to ascertain information about the structure of nuclei. Comparing the binding energies of nuclides that differ only in the number of neutrons may give insights into how nucleons are arranged within nuclei. Previous work indicates that when two neutrons and two protons are present in a nucleus they form an alpha particle. Thus the addition of a neutron (or proton) that completes the components necessary for the formation of an alpha particle has a special effect quite different from the addition of a neutron where there is no effect associated with alpha particle formation. In the following analysis the effect of additional neutrons is analyzed by starting with a nuclide that could (and in all probability, does) contain an integral number of alpha particles, hereafter referred to as alpha nuclides, and examining the binding energy of nuclides with successively more neutrons.

For example, consider the case of the Selenium 68 nuclide. This nuclide could contain 17 alpha particles. Its binding energy is 576.4 million electron volts (MeV). The Selenium 69 nuclide has a binding energy of 586.62 MeV, so the effect of the additional neutron is an increase in binding energy of 10.22 MeV. The graph of the additional binding energy as a function of the number of additional neutrons is as follows.

The effect of additional neutrons on binding energy is very regular. It appears that it could be closely approximated as a parabola. The parameters of the parabola can be obtained by fitting a quadratic function to the data using multiple regression analysis. However more insights into the nature of the relationship between binding energy and additional neutrons can be obtained by examing the incremental increases in binding energy as successive neutrons are added. The graph of these incremental increases is very interesting.

The alternating pattern of smaller then larger increases is undoubtably associated with the formation of spin-pairs of neutrons. (The use of the term spin is metaphorically and may not be the same as what is meant by spin macroscopically.)

As the graph above shows, when the alternating pattern is ignored the successive increments involve a diminishing level. However there qualitative change in the pattern after 16 or 17 neutrons have been added. There is a notable decrease in the amplitude of the alternating fluctuations. The amplitude of the alternating fluctuations is approximately constant from 1 up to 16 and then is approximately constant at a lower level from 16 up to 24. This break in the amplitude of the fluctuations would seem to be important information on the structure of the arrangement of neutrons in nuclei.

There is further analysis for the Selenium 68 nuclide in another study, but here it is appropriate to look for the same phenomena for the addition of neutrons to other alpha nuclides. The case for tin is of special interest because tin has the largest number of stable isotopes. The alpha nuclide Sn 100 nuclide sustains the addition of 37 neutrons. The graph of the incremental increases of binding energy for these nuclides is shown below.

As with Selenium there are alternating fluctuations about a downward trend until certain number of additional neutrons is reached. In the case of tin this break in the pattern occurs at 32 neutrons instead of the 16 in the case of selenium. In the case of tin there is a drop both in the level of the incremental increases and the amplitude of the alternation.

For the case of strontium there appears to be a change to a reduced amplitude of fluctuation over a range and then an increase in the amplitude. The range of reduced amplitude covers the addition of about eight neutrons. The first break in the pattern occurs at 13 neutrons and the second at about 21 neutrons.

The pattern for krypton is about the same as for strontium. The first break in the pattern occurs for 14 neutrons and the second for 21.

For other elements the same pattern seems to be there but obscured by there being a more limited range of the number of neutrons which may be added and a stable nuclide obtained, for example the cases of zinc, titanium and calcium.

For other cases such as iron and nickel the pattern is harder to identify.

However after having seen the pattern for heavier nuclides it is possible to observe it on a more limited scale for the lighter nuclides such carbon and oxygen.

Even for the cases of helium and berylium the pattern can be discerned.


Generally there is a parabolic relationship between the increase in binding energy and the number of additional neutrons. That is to say, the binding energy goes up as neutrons are added but the incremental increase declines with the number of additional neutrons. However the incremental increases fluctuate between smalled increases and larger increases. At some number of additional neutrons there is a break in the relationship that may involve a downward shift in the level of the increments and/or a change in the amplitude of the fluctuations. For a more detailed quantitative analysis see Quantitative Impact of Additional Neutrons.

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