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The Second Differences of Binding Energies Is Equal
to the Interaction Energy of the Last Neutron
with the Next to Last Neutron 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 neutrons and two protons 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 neutron for nuclides containing the same number of protons but varying numbers of neutrons can be tabulated. Likewise such a tabulation can be created for nuclides containing the same number of neutrons but varying numbers of protons.

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 neutrons and protons, is given in the following table.

Shell
Number
12345678
Capacity2481422324458
Range1 to 23 to 67 to 1415 to 2829 to 5051 to 8283 to 126127 to 184

The plot of the incremental binding energy of the 20th neutron versus the number of protons in the nuclide is shown below.

The graph shows neatly the magicality of 20 and 28 where the IBE drops sharply. A sharp drop in the IBE is an indication that a shell or subshell has been filled and additional nucleons are going into a higher shell. The sawtooth pattern is due to the energy involved in the formation of spin pairs.

The Nature of the Incremental Binding Energy of the Neutrons

The increase in the incremental binding energies of a neutron as a result of an increase in the number of neutrons is equal to the interaction of the last neutron with the next to last last neutron, provided these two are in the same neutron shell.

Rationale:
Consider a nuclide with n neutrons and p protons. The binding energy of that nuclide represents the net sum of the interaction energies of all n neutrons with each other, all p protons with each other and all np interactions of neutrons with protons. Below is a schematic depiction of the interactions.

  

The black squares are to indicate that there is no interaction of a neutron with itself. The diagram might seem to suggest a double counting of the interactions but that is not the case.

The neutron incremental binding energy is the difference in the binding energy of the nuclide with n neutrons and p protons and that of the nuclide with n-1 neutrons and p protons. In the diagrams below the interactions for the nuclide with (n-1) neutrons and p protons are colored.

  

That subtraction eliminates all the interactions of the p protons with each other. It also eliminates the interactions of the n-1 neutrons with each other and the n-1 neutrons with the p protons. What is left is the interaction of the n-th neutron with the other n-1 neutrons and the interaction of the n-th neutron with the p protons.

Now consider the difference of the IBE for n neutrons and p protons and the IBE for (n-1) neutrons and p protons. These are shown as the white squares in the diagrams below. The colored squares are the interactions for the IBE of a neutron in a nuclide of (n-1) neutrons and p protons.

 

The subtraction of the IBE for (n-1) neutrons and p protons from the IBE for n neutrons and p protons depends upon the magnitude of the interaction of the (n-1)-th neutron with the different neutrons compared to the interaction of the n-th neutron with those same neutrons. Visually this is the subtraction the values in the green squares from the white squares on the same level. When the n-th and the (n-1)-th neutrons are in the same shell the magnitude of the interactions with any other neutron are, to the first order of approximation, equal. Thus the interactions with the p protons are entirely eliminated. Likewise for the first (n-2) neutrons. All that is left is the interaction of the n-th neutron with the (n-1)-th neutron.

Note that the interaction of the n-th and (n-1)-th neutrons may or may not involve the interaction associated with the formation of a neutron spin pair. The graph below illustrates the effect of spin pair formation.

The Separation of the Interaction Energy of
Successive Neutrons into Two Components

Within a shell the average of the IBE for the n-th and (n+2)-th neutron give a good approximation of what the IBE would be if there were no spin pair formed.

The difference between the actual IBE and the average of the adjacent figures is a good approximation of the energy involved in the pair formation. The change in the IBE due to the nuclear strong force per additional neutron is computed as one half of the change in IBE between the value for the n-th neutron and the value for the (n-2)-th neutron. Its value is always negative, indicating that the nuclear strong force between two neutrons is a repulsion.

The decline and then rise in the values is associated with the filling of the fourth neutron shell and the passage to the fifth shell at 28 neutrons. The pattern with a single shell is approximately linear.

Below are some illustration of the relationship for other proton numbers

In this case the downward spike is associated with the magic number 28.

In this case the downward spike is associated with the magic number 82.

In this case the downward spike is associated with the magic number 126.

A Comparison of the Interaction Due to the Strong Force of Different Numbers of Protons with Neutrons in the Same Shell

In order to avoid the effect of spin pair formation it is necessary to compare two odd number or two even numbers of protons.

The values for 25 and 27 protons differ but there is a distinct tendency over a range of neutron numbers for the displays to be at the same general level.

The Sensitivity of the Results to Possible Errors
in the Conventional Mass of the Neutron

The binding energy of all nuclides are computed as the energy value of its mass deficit. The mass deficit of all nuclides except one are computed as the difference between the mass of their constituent neutrons and protons and the mass of the nuclide. The mass of any charged particle can be measured by injecting it into a magnetic field and measuring the radius of the orbit it makes. The mass of a neutron, since it is a neutral particle, cannot be measured in this way. Instead its mass is deduced from the masses of a deuteron and a proton and an estimate of the mass deficit of the the deuteron. When a deutron is formed a gamma photon of energy of 2.25 million electron volts is emitted. This is taken to be the mass deficit of the deuteron. This may not be the correct value for the mass deficit of the deuteron.

If the mass of the neutron is in error by an energy amount Δ then the binding energy of any nuclide with n neutrons is in error by nΔ. The incremental binding energy of a nuclide with n neutrons is the difference between its binding energy and that of the nuclide with the same number of protons but (n-1) neutrons. Thus the incremental binding energy is in error by an amount Δ.

Thus, any difference in incremental binding energies is
unaffected by any error in the mass of the neutron.

Thus the magnitude of the second differences in binding energy are meaningful and their signs will not be altered by a correction in the mass of the neutron.

Conclusion

The conclusion to be drawn is that the second differences with respect to the number of neutrons are a linear function of the number of neutrons in the shell up to the point of near complete filling of the shell. The second differences represent the interaction of a neutron with the previous one in the nuclide. The interaction of the last two neutrons is almost always negative thus indicating a repulsion between them.


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