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Illustrations of the Effect of Double Pairing
on the Incremental Binding Energy of Neutrons

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 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. The diagrams below depict the situation.

 

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. 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 spin pair.

Empirical Tests

The plot of the incremental binding energies of the isotopes of Strontium (proton number 38) versus the number of neutrons in the nuclide is shown below.

The graph shows neatly the magicality of 50 where the IBE drops sharply as a result of a neutron shell being filled and the next neutron going into a higher shell. The sawtooth pattern is due to the energy involved in the formation of spin pairs of neutrons. The sharp drop after 38 neutrons might appear to indicate some magicality to 38, perhaps a subshell, but such is not the case. When the number of neutrons is less than the number of protons each additional neutron results in the formation of a neutron-proton pair. Beyond the point where the number of neutrons equals the number of protons no pair can be formed. The level of the IBE drops. If the proton number is even then the drop in IBE due to no neutron-proton pair being formed combines with the drop due to no neutron-neutron pair being formed to produce a notably sharp drop in the IBE for the next neutron number.

Here is the graph for the case of the isotopes of Krypton (proton number 36).

Here again there is a sharp drop when the number of neutrons exceeds the proton number of 36.

The case of an odd proton number is of interest. Here is the graph for the isotopes of Rubidium (proton number 37).

The addition of the 38th neutron brings the effect of a neutron-neutron pair but a neutron-proton pair is not formed, as was the case up to and including the 37th neutron. The effects almost but not exactly cancel each out. It is notable that the binding energy involved in the two types of nucleonic pairs are almost exactly the same.

This same pattern is seen in the case for the isotopes of Bromine.

The values of the IBE of neutrons within a particlulr shell for a particular proton number can be explained by an equation of the form

IBE = c0 + c1u(n≤p) + c2m + c3e

where m is the number of neutrons in the shell (n-28), u(n≤p) is 1 if n≤p and 0 otherwise and e is 1 if n is even and 0 otherwise.

The regression equation for the isotopes of Bromine with neutron numbers in the fifth shell (n=29 to 50) is

IBE = 13.15159 + 2.22057u(n≤35) − 0.299396m + 2.27264e
[13.6] [-25.7] [24.4]

The coefficient of the determination for this equation is 0.99455. This means that almost 99.5 percent of the variation in the IBE of those isotopes of Bromine are explained by variations in the three variables. The coefficient of e, 2.27264 MeV, is the binding energy accounted for by the formation of a neutron-neutron pair. The coefficient of u(n≤35), 2.22057 MeV, is the binding energy accounted for by the formation of a neutron-proton pair. The values are close but not equal. The coefficient of m, −0.299396 MeV, is the binding energy resulting from the interaction of successive neutrons. The fact that it is negative indicates that the force between two neutrons is a repulsion.

The regression equation for the isotopes of Krypton with neutron numbers in the fifth shell (n=29 to 50) is

IBE = 13.2774 + 2.06919u(n≤36) − 0.30390m + 3.17100e
[13.2] [-25.1] [35.3]

The coefficient of determination for this equation is slightly better than the one for the isotopes of Bromine, 0.99649. The big surprise is that the binding energy due to a neutron-neutron pair formation, 3.17100 MeV, is more than 50 percent greater than the one for a neutron-proton pair formation, 2.06919 MeV.


For material on the case for the cross differences of binding energy being the interaction energy of the last nucleon with the last nucleon of the opposite type see Cross Differences. For material on the differences in the incremental binding energies of neutrons see Neutrons.


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