San José State University

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A Statistical Regression Equation for the
Binding Energies of Nuclei Taking Into Account
the Heteroskedasticity of the Random Term

Although nuclei are composed of neutrons and protons these nucleons are further organized into structures; neutron spin pairs, proton spin pairs, and neutron-proton spin pairs. In addition to these there are linkages of neutrons and protons of the form -n-p-p-n-, or equivalently, -p-n-n-p- that induce binding energy effects similar to alpha particles. These linkages are called alpha modules in the following analysis. In addition to alpha modules and the three types of spin pairs a nucleus may contain singleton neutrons or protons. These latter are not substructures and their formation do not contribute to the binding energy of the nucleus they are in.

Binding energy is also determined by the interactions of the various substructures but the analysis below presumes that the interactions of the substructures reduces down to interactions among neutrons and protons.

The notation which is used is

The number of interactions of neutrons with each other is N(N-1)/2 and likewise for proton interactions. The number of separate interactions of neutrons with protons NP. The binding energy BE is then assumed to be a linear homogeneous function of the numbers of substructures and the numbers of interactions.

The regression equation based upon the above is

BE = 39.01#a + 13.64#nn + 1.80#pp + 6.58#np
−0.188(N(N-1)/2) −0.486(P(P-1)/2) + 0.273NP

t-ratios
[510.0]   [76.8]   [3.2]   [14.2]  
[-36.9]   [-41.2]   [35.8]

R² = 0.99989

The standard error of the estimate for this equation is 12.65 MeV. The mean value and standard deviation of the binding energy for the 2931 nuclides are is 1071.9 and 504.9 MeV, respectively. The coefficient of variation for the equation is 12.65/1071.9 or 1.2 percent. This seems to be good, but a margin of error of 12.65 MeV at the level of the average binding energy but a margin of error of 12.65 MeV at the level of the smallest nuclides where the binding energy is on the order of 8 MeV or less is a different matter. The coefficient of variation of the regression equation for the smaller nuclides is on the order of 100 percent.

What is needed is a method of statistical estimation that gives more weight to the smaller nuclides. The standard statistical is in the nature of the following.

z = ax + by + u
where x, y and z are variables
and a and b are parameters.

The stochastic variable u is presumed to have a constant standard deviation σ. The situation relevant for the binding energy data is that σ is not constant. A strong possibility is that

σ = zσ0

where z=ax+by. Generally z is not known, but since z=z+u, z be used as a good approximation of z. Thus the equation

z = ax + by + u
may be divided by z
to give
1 = a(x/z) + b(y/z) + u/z

The stochastic variable u/z has approximately constant standard deviation.

When this sort of scheme is applied to the binding energies the regression equation which results is

1 = 40.272(#a/BE) + 0.907(#nn/BE) + 2.269(#pp/BE) + 2.157(#np/BE)
− 0.289N(N-1)/(2BE) −0.887 P(P-1)/(2BE) + 0.504(NP/BE)
t-ratios
[351.4] [9.3] [9.4] [16.9]
[-14.8] [-24.8] [19.2]

R² = 0.99642

The standard error of the estimate is 0.06 and this is also the coefficient of variation for the regression.

The equation for predicting the binding energy of a nuclide on the basis of its composition is

BE = 40.272#a + 0.907#nn + 2.269#pp + 2.157#np
− 0.289N(N-1)/2 −0.887 P(P-1)/2 + 0.504NP

Although it is hard to rationalize the magnitudes of some of the coefficients this equation appears to be superior to the equation in which heteroskedasticity was not taken into account. Estimates for some of the parameters are available from the binding energies of small nuclides. For example, the binding energy of a deutron, essentially a neutron-proton spin pair, is 2.22457 MeV. This is comparable to the coefficient of 2.157 MeV found for #np. The binding energy of a deuteron covers the strong force interaction of a neutron and a proton as well as the formation of a neutron-proton spin pair. An alpha particle is a special case of an alpha module and its binding energy is 28.295674 MeV, far less than the value of 40.272 MeV found in the regression.

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