San José State University

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 The Functional Form of the Incremental Binding Energy of Neutrons as a Function of the Number of Neutrons in the Nuclide

## Mass Deficits and Binding Energies

The mass of a nuclide, such as helium 4, the alpha particle, is less than the masses of the two protons and two neutrons of which it is composed. The difference is called the mass deficit and that mass deficit expressed in energy units via the Einstein formula E=mc² is called the binding energy. The binding energies have been measured for almost three thousand nuclides.

## Incremental Binding Energies

The incremental binding energy (IBE) of a nucleon (neutron or proton) in a nuclide is the difference in the binding energy of that nuclide and the nuclide containing one less nucleon of the same type.

The incremental binding energies of a proton for nuclides containing the same number of neutrons but varying numbers of protons can be tabulated. Likewise such a tabulation can be created for nuclides containing the same number of protons but varying numbers of neutrons.

## Nucleonic Shells

Protons and neutrons are arranged separately in shells. The numbers corresponding to the shells filled to full capacity are known as the nuclear magic numbers. Conventionally the magic numbers are {2, 8, 20, 28, 50, 82, 126}, but a case can be made for the magic numbers being instead {2, 6, 14, 28, 50, 82, 126} with 8 and 20 being in a different category of magic numbers. For more on this see Magic Numbers.

The structure of the nuclear shells, both for protons and neutrons, is given in the following table.

 ShellNumber 1 2 3 4 5 6 7 8 Capacity 2 4 8 14 22 32 44 58 Range 1 to 2 3 to 6 7 to 14 15 to 28 29 to 50 51 to 82 83 to 126 127 to 184

The plot of the incremental binding energy of the neutrons in the nuclides with proton number 26, the isotopes of iron, versus the number of neutrons in the nuclide is shown below.

The sawtooth pattern is due to the energy involved in the formation of spin pairs. It is desirable to separate the effect of spin pair formation from the effect due to the interaction of nucleons through the nuclear strong force. One way of doing that is shown below.

However even with this modified method of computing the incremental binding energies the graph displays a sawtooth pattern. An alternate approach is to plot the data for only the even values of the number of neutrons. This is shown below.

The curve is relatively smooth. The increments in the incremental binding energies reveal more details.

The sharp downward spike is associated with the neutron number being equal to the proton number of 26. The drop comes after 26 so it is labeled with 28.

A regression equation in which the explanatory variables are the amount n is below p, the amount n is above p and whether or not n is equal to p explains 88 percent of the variation in the increments in the incremental binding energies. If the increments in the incremental binding energies are one linear function for n below p and another linear function for n above p then the incremental binding energies are one quadratic function for n below p and another quadratic function for n above p. A regression equation of this form explains 99.4 percent of the variation in the incremental binding energies.

The corresponding cases for Manganese and Cobalt are shown below.

When the incremental binding energies for the three above cases are plotted togeither in the same graph versus not the number of neutrons but instead the difference between the number of neutrons and the number of protons the result is a surprising near equality of the patterns.

For the incremental binding energy to be a function of (n-p) means that the interaction of a neutron and a proton on another neutron is opposite in effect. That is say neutrons are attracted to protons but repelled by other neutrons.

(To be continued.)

## Conclusion

The conclusions based upont the limited sample considered is that the general form of the increments of the incremental binding energy is

This means that the functional form for the incremental binding energy due to the interaction of the nucleons through the strong force is