San José State University

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The Statistical Explanation of the Binding Energies
of Nuclides in terms of their Pair Formations and the
Interactions of their Nucleons through the Strong Force

Background

The nucleons (protons and neutrons) of a nucleus, whenever possible, form spin pairs (neutron-neutron, proton-proton and neutron-proton). Such spin pair formation is exclusive in the sense that one nucleon can form a spin pair with one other nucleon of the same type and with one nucleon of the opposite type. A nucleon can have interaction with any number of other nucleons through the so-called nuclear strong force . The term strong is inappropriate because that force is not all that strong compared to the forces involved in spin pairing. A more appropriate name for the so-called strong force is the nucleonic force, the force between nucleons.

A neutron spin pair and a proton spin pair can form an alpha particle whose binding energy is significantly greater than the sum of the binding energies due to the spin pair formations within it. More generally the nucleons are linked together in chains containing sequences of the form -n-p-p-n-, or equivalently -p-n-n-p-, which will be called alphaa modules. The chains of alpha modules form rings in shells. The lowest shell is just an alpha particle.

The binding energy of a nuclide is expressed in millions of electron volts (MeV). The data for the nuclides were tabulated in terms of their numbers of alpha modules and the number of spin pairs outside of the alpha modules. Nuclides may also contain a singleton (unpaired) neutron or proton. These do not involve any contribution to binding energy due to pairing and they are left out of the analysis for now but will be reconsidered later.

The binding energy of a nuclide is also affected by the interaction through the nucleonic force of the nucleons. \ If n and p are the numbers of neutrons and protons, respectively, the number of interactions of the three types are ½n(n-1), np and ½p(p-1).

The Regressions

The regression of the binding energies of the 2931 nuclides on the numbers of alpha modules and other spin pairs and on the numbers of nucleonic force interactions gives the following.

Regression Results
Variable Coefficient
(MeV)
t-Ratio
α 42.07905 764.2
nn 13.89456 151.9
pp 14.50004 44.6
np 12.62388 44.1
p(p-1)/2 -0.54606 -87.1
np 0.29681 73.5
n(n-1)/2 -0.20415 -75.6
Const. -44.40578 -85.9

The coefficient of determination (R²) for this equation is 0.99982 and the standard error of the estimate is 6.7 MeV. The average of the binding energies is 1071.9 MeV so the coefficient of variation for the regression equation is 0.625 of 1 percent.

The t-ratio for a coefficient is the ratio of its value to its standard deviation. The magnitude of the t-ratio must be two or greater for the coefficient to be statistically significantly different from zero at the 95 percent level of confidence. As can be seen the values of the t-ratios indicate that the likelihood that their values are due solely to chance is infinitesimally small.

All of the coefficients for the spin pairs are positive indicating the associated force is attractive. They are also approximately of the same magnitude, roughly 14 MeV.

The coefficients for the interaction of nucleons through the so-called strong force are especially interesting. The coefficients for the interactions of like nucleons are both negative indicating that the forces between like nucleons are repulsions The coefficient for the interaction of unlike nucleons is positive, indicating that the force between unlike nucleons is an attraction.

The force between like nucleons being a repulsion and being an attraction between unlike nucleons is explained by protons and neutrons having nucleonic charges. The force between two nucleons is proportional to the product of their nucleonic charges. The charges of the neutron and proton differ in sign. Thus if two nucleons are alike the product of their charges is positive and the force is a repulsion; if they are unlike the sign of the product is negative and the force is an attraction.

If the nucleonic charge of a proton is taken to be 1 and that of a neutron is denoted as q then the interaction between two neutrons is proportional to q² whereas that between a neutron and a proton is proportional to q. Thus the ratio of the interaction between neutrons to the interaction of a neutron and proton should be equal to q. The estimates of the interactions in the above table give

q = −0.20415/0.29681 = −0.678658274

Since q is most likely the ratio of small integers this means that q is equal to −2/3.

The interaction between protons is complicated by the effect of the electrostatic repulsion. The interaction between protons is proportional to (1+d) rather than 1. Thus the ratio of the interaction between a neutron and a proton and the interaction between protons is proportional to q/(1+d). From the above table that ratio is −0.54354. The value of d has been estimated in another study to be about 0.2. Thus the value of q is equal to −0.54354*(1.2), which is −0.65, again essentially −2/3.

If the ratio of the interaction of neutrons to the interaction of protons, which should be proportional to q²/(1+d), is used the estimate of |q| is 0.664.

Singleton Nucleons

A singleton neutron or a singleton proton does not account for any binding energy due to spin pairing but there could be an effect on binding energy due to a rearrangement\ or adjustment in the structure of the nucleus that their presence makes possible. Such adjustments affect the interactions among the other nucleons through the strong force. The regression including the singleton nucleons gives the following results.

Additional Regression Results
Variable Coefficient
(MeV)
t-Ratio
α 42.42517 992.8
nn 13.29281 185.4
pp 15.05737 61.7
np 18.9597 72.1
1n 5.97161 22.6
1p 12.74778 47.7
p(p-1)/2 -0.57266 -119.3
np 0.31040 101.1
n(n-1)/2 -0.21065 -103.3
Const. -51.97957 -123.1

The coefficient of determination (R²) for this equation is 0.99990 and the standard error of the estimate is 5.0 MeV. Since the average of the binding energies is 1071.9 MeV so the coefficient of variation for the regression equation is 0.47 of 1 percent.

The magnitudes of the coefficients are approximately the same as those in the regression without the effect of the singleton nucleons. The estimate of q, the nucleonic charge of a neutron relative to that of a proton, is −0.21065/0.31040=−0.68781, essentially −2/3.

As can be seen in the table the effects of the singleton nucleons are statistically significant and relatively large. On average a singleton neutron is interacting with 56 protons and 77.4 neutrons. Applying the interaction binding energies of +0.31040 MeV and −0.21065 MeV accounts for only 0.75 MeV of the value of 5.97 MeV found. Thus another phenomenon is involved and that is the adjustment in the structure of the other nucleons that results from the presents of a singleton nucleon.

Conclusions

The model is essentially that the binding energy of a nuclide is the sum of the binding energy associated with the formation of nucleon pairs including a special effect for alpha modules and the binding energy due to the interaction of the nucleons through what conventionally but inappropriately is called the strong force. It accounts for 99.99 percent of the variation in the binding energies of the 2931 nuclides when the effects of singleton nucleons are included.

The results are consistent with the nucleonic charge of a neutron relative to that of a proton being −2/3.


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