San José State University

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The Binding Energetics of Nuclei
with 12 Protons or Neutrons

Dedicated to Chuck
A friend for sixty years
and an always willing reader

The Nature of the Nucleonic
Substructures of Nuclei

The mass of a nucleus containing n neutrons and p protons is less than the combined masses of n neutrons and p protons. The difference is called the mass deficit of the nuclide. When the mass deficit is expressed in energy units via the Einstein formula E=mc² the value is called the binding energy of the nuclide.

If BE(n, p) is the binding energy of a nuclide with n neutrons and p protons then the incremental binding energy IBEN of a neutron is

IBEN(n, p) = BE(n, p) − BE(n-1, p)

The incremental binding energy of a proton, IBEP, is

IBEN(n, p) = BE(n, p) − BE(n, p-1)

Incremental binding energies can reflect the formation of any of the three types of spin pairs, neutron-neutron, proton-proton and neutron-proton. They also include the effect of the interaction of the additional nucleon with the nucleon already in the nuclide. The magnitudes involved can be illustrated with the data for magnesium isotopes. These are shown below in graphic and tabular form.

The Incremental Binding Energies of Neutrons

When the number of neutrons is below 12 the addition of another neutron will result in the formation of a neutron-proton pair. The incremental binding energy for all levels of the number of neutrons shows the net binding energy due to the interaction of the additional neutron with the other nucleons in the nuclide. The binding energy due to the formation of a neutron-neutron spin pair can be estimated by comparing the binding energy for a case in which a neutron-neutron spin pair is formed with the average of the IBE for the adjacent IBE's. Those values are shown in the following table. The values for the cases of twelve or less neutrons may also show the effect of the formation of a pair of neutrons with a pair of protons, which may be called an alpha module. There is too little data to tell definitely.

The Binding Energy Quantities
for the Isotopes of Magnesium
(12 protons)
Neutrons Binding
Energy
(MeV)
IBE
Incremental
Binding
Energy
(MeV)
Average of
Adjacent
IBE
(MeV)
Binding
Energy
Due to
n-n Pair
Formation
8 134.47
9 149.198 14.728
10 168.5776 19.3796 13.9376 5.442
11 181.7248 13.1472 17.955845 4.808645
12 198.25689 16.53209 10.238935
13 205.58756 7.33067 13.81258
14 216.68063 11.09307 6.88701 4.20606
15 223.12398 6.44335 9.798295 3.354945
16 231.6275 8.50352 5.077925 3.425595
17 235.34 3.7125 7.39676 3.68426
18 241.63 6.29 3.06125 3.22875
19 244.04 2.41 5.97 3.56
20 249.69 5.65 2.24 3.41
21 251.76 2.07 5.245 3.175
22 256.6 4.84 1.135 3.705
23 256.8 0.2 4.17 3.97
24 260.3 3.5 0.05 3.45
25 260.2 -0.1

The plot of the data on the binding energy due to the formation of a neutron-neutron spin pair, shown below, indicates that for more than twelve neutrons that value is between 3 and 4 MeV. The average for 15 to 24 neutrons is 3.56 MeV.

The difference between the binding energy due to pair formation for the nuclides with 12 or fewer nuclides and that for nuclides with more than 12 neutrons is due to another factor. There is a transition to a higher shell at the magic number of 14. It is not the binding energy due to the formation of a neutron-proton pair.

When the effect of the formation of a neutron-neutron spin pair is subtracted from the incremental binding energy the result includes the binding energy due to the interaction through the nuclear strong force, but for the cases of the neutron number of 12 or below it includes the binding energy due to the formation of a neutron-proton pair. Those values are shown in the following graph.

The significant aspect of the display is that the more neutrons there are in the nuclide the smaller is the effect. This is an indication that the force between two neutrons is repulsion.

There appears to be a drop in the value after n=12. The value of that drop is the binding energy involved in the formation of a neutron-proton pair. Its value can be estimated by the use of the following regression equation

IBEN = c0 + c1n + c2m

where m is equal to 1 if n is less than or equal to 12 and 0 otherwise.

The results of the regression are;

IBEN = 16.184 − 0.694n + 4.648m
                     [-16.4]   [9.4]

The numbers in brackets below the regression coefficients are their t-ratios; i.e., the ratio of the coefficient to its standard deviation. The t-ratios indicate that the coefficients are statistically significantly different from 0. The coefficient of determination (R²) is 0.992.

Thus the estimated binding energy associated with the formation of a neutron-proton spin pair is 4.648 MeV.

The constant in the equation, 16.184 MeV, represents the effect if the number of neutrons were zero. This would be the effect of the 12 protons. That is 1.349 MeV of strong force interaction per proton. The estimated strong force interaction of a neutron with another neutron is −0.694 MeV.

The relative magnitudes of the effects on binding energy are significant. From the above it seems that the formation of neutron-neutron spin pair results in an increase in binding energy of 3.56 MeV. The net binding energy due to the interaction of the additional neutron with 12 protons and 14 neutrons for example is 6.887 MeV. This is only 0.265 MeV per strong force interaction. This is why for small nuclides the energetics are so dominated by the effects of spin pair formation. For larger nuclides the large number of interactions through the strong force becomes overwhelmingly large compared to the effect of spin pair formation from the additional neutron.

The figure of 0.265 MeV per strong force interaction is a net figure of the positive value for interactions of the additional neutron with the protons of the nuclide less the negative effect of the strong force interaction of the additional proton with the number of neutrons present in the nuclide.

The regression analysis separated the effects of the protons and the neutrons on the additional neutron. As indicated previously the binding energy effect of the strong force of a proton on a neutron is 1.349 MeV whereas that of a neutron on a neutron is −0.694 MeV. The ratio of these two values is −0.515. Other work puts the ratio of these binding energy effects at −2/3. This is a reasonable confirmation that the effects of a neutron on a neutron is of the opposite sign and smaller in magnitude than the effect of a proton on a neutron.

The Incremental Binding Energies of Protons

Generally the picture of the relationships for protons is the same as for neutrons.

The Binding Energy Quantities for the Nuclides with 12 neutrons
Protons IBE
Incremental
Binding
Energy
(MeV)
Average of
Adjacent
IBE
(MeV)
Binding
Energy
Due to
p-p Pair
Formation
6 26.09
7 16.348 22.72135 6.37335
8 19.3527 13.74075 5.61195
9 11.1335 17.309205 6.175705
10 15.26571 9.963805 5.301905
11 8.79411 13.47929 4.68518
12 11.69287
13 2.27131 8.605335
14 5.5178 1.582655 3.935145
15 0.894 3.9939 3.0999
16 2.47 -0.443 2.913
17 -1.78 1.42 3.2
18 0.37 -0.89

There is a transition to a higher proton shell at the magic number of 14. This shows up as a difference in the level of the estimate above 14 protons compared to the values for 14 and below.

The average binding energy for the formation of a proton-proton spin pair for 15 to 17 protons is 3.071 MeV, roughly the same as 3.56 MeV for the formation of a neutron-neutron spin pair. When the effect of the formation of a proton-proton spin pair is subtracted from the incremental binding energy the result includes the binding energy due to the interaction through the nuclear strong force, but for the cases of the proton number of 12 or below it includes the binding energy due to the formation of a neutron-proton pair. Those values are shown in the following graph.

The plot of the binding energies due to the interaction of the additional proton indicate the more protons in the nuclide the smaller interaction binding energy, thus indicating that the force between protons is repulsion.

The value of that drop after p=12 is the binding energy involved in the formation of a neutron-proton pair. Its value can be estimated by the use of the following regression equation

IBEP = c0 + c1p + c2q

where q is equal to 1 if p is less than or equal to 12 and 0 otherwise.

The results of the regression are;

IBEN = 25.782 − 1,659p + 1.12q
              [-6.3]   [0.6]

The numbes in brackets below the regression coefficients are their t-ratios; i.e., the ratio of the coefficient to its standard deviation. The t-ratios indicate that the coefficient of p is statistically significantly different from 0 at the 95 percent level of confidence but that of q is not. The coefficient of determination (R²) of the equation is 0.990.

The constant is the extrapolated effect with no protons and hence is the effect of the 12 neutrons. This is 2.148 MeV per neutron. The effect of a proton on a proton is, in this regression, −1.659. The negative sign indicates the strong force between two protons is repulsion.

Conclusions

The incremental binding energies of neutrons and of protons provides a way to estimate the binding energies due to the formation of the three types of spin pairs, neutron-neutron, proton-proton and neutron-proton. The binding energy due to the strong force interactions of the nucleons can be estimated and regression analysis used to separate it into its components.

(To be continued.)

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