San José State University

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 An Alpha Particle Version of the Nuclear Shell Model and Its Statistical Performance Explaining Binding Energies

There is significant evidence that the protons and neutrons within a nucleus are, to the extent possible, organized into alpha particles. If this is the case then the mass deficits (binding energies) of nuclides are the result of three processes. First, there is the creation of alpha particles, each of which has a binding energy of about 28.3 MeV. Second, there is the binding energies resulting from the organizational arrangement of the alpha particles in the nuclides. Third there is the binding energy resulting from the fitting of the extra protons and neutrons, if any, into the arrangement of alpha particles.

If the nucleons in a nucleus are arranged into alpha particles then the conventional figures on binding energies misrepresent the stability of the nuclides. The energy it takes to break a brick in half is entirely different from the energy that it would take to pulverize the brick into dust.

Binding Energies

There is information on the binding energies of 2931 nuclides, including the proton and the neutron. The binding energies range from −0.76 MeV for Be5 to 1978.4 MeV for an artificially created nuclide. The average binding energy is about 1072 MeV. The average number of protons for the 2931 nuclides is 56 and the average number of neutrons is 78.2. The sum of the squared deviations from the average is called the variance or variation. The square root of the variance is called the standard deviation. For the binding energies the standard deviation is 504.8 MeV. Total Variance is 254,825.2 (MeV)².

The binding energy of an alpha particle is about 28.3 MeV. If the possible number of alpha particles contained in a nuclide is multiplied by the 28.3 MeV figure per alpha particle and then this figure is subtracted from the binding energy the result, which will be called excess binding energy, binding energy due to the arrangement of alpha particles. The average level of this excess binding energy is 289.4 MeV and its standard deviation is is 138.95 MeV. The variance of the excess binding energies is 19,306.7 (MeV)². When this figure is compared with the figure of 254,825.2 (MeV)² it is revealed that 92.4 percent of the variation in binding energy is explained as being due solely to the binding energies of the alpha particles they contain.

The Explanation of the Excess Binding Energies of the Nuclides

If the excess binding energies of the 25 nuclides which could contain an integral number of alpha particles are plotted versus the number of alpha particles the results are as shown below.

The excess binding energies of the nuclides which could contain an integral number of alpha particles plus four neutrons are plotted versus the number of alpha particles the relationship shows a bend. A similar sort of thing occurs for 41 alpha particles.

This suggests a regression equation of the form

XSBE = c1#α + c2u(#α−2) + c3u(#α−14) + c4u(#α−25) + c5u(#α−41)

where u(#α)-K is a function that is 0 is #α≤ K and #α-K otherwise. The function u(z) is sometimes called the ramp function.

When the regression is carried out the result are:

XSBE = 9.31772#α + 3.20956u(#α−2) − 0.90191u(#α−14) −5.21067u(#α−25) − 0.21399u(#α−41)

The coefficient of determination, R², for this regression is 0.74435. This means that 74.435 percent of the 7.6 percent of the variation not explained by the binding energies of the constituent alpha particles. Altogether the explained variation in binding energies is 92.4+(0.74435)(7.6)=98.06 percent.

The regression coefficients are the incremental changes in the slopes. The cumulative values that represent the slopes of the relationship

XSBE = 9.31772#α + 12.52728u(#α−2) + 11.62536u(#α−14) + 6.41469u(#α−25) + 6.20070u(#α−41)

If the 28.3 MeV is added to each of these the result is a equation for binding energies:

XSBE = 37.61339#α + 40.82295u(#α−2) + 39.92104u(#α−14) + 34.71036u(#α−25) + 34.49637u(#α−41)

The statistical performance of explaining 98 percent of the variation is not bad but the model is intended as a minimalist one to use for comparison to more sophisticated models.

The t-ratios for the coefficients except those for #α and the #α greater than 25 and less than 41 are not significantly different from zero at the 95 percent level of confidence. This means that the other variables may be dropped from the regression without much loss of explanatory power. That regression is

XSBE = 11.89347#α − 5.55242u(#α−25)

The coefficient of determination for this regression (R²) is 0.744257, virtually the same as the regression with all of the variables.

The inclusion of a constant term in this regression and the one with all of the variables did not produce a value that was significantly different from zero at the 95 percent level of confidence.

The t-ratio for the coefficient for #α is 52.7 and for u(#α-25) it is -15.4. The regression indicates that for each additional alpha particle in a nuclide up to 25 the increase in the excess binding energy is 11.89347 MeV. Beyond 25 the increase is 6.34106 MeV. The total including the binding energy for the alpha particle itself is 40.18915 Mev up to 25 alpha particles and 34.63673 MeV beyond 25.

A regression using only #α performs nearly as good. The coefficient of determination (R²) is 0.6952. The regression coefficient is 10.48174 MeV, indicating that for each addition alpha particle in a nuclide the binding energy increases by 38.77741 MeV.

Conclusion

A statistical model for explaining the binding energies of nuclides has to have a coefficient determination above 98 percent to be considered relevant because the simplest alpha particle model achieves a statistical performance at that level. The effect of an additional alpha particle on binding energy is in the range of 35 to 40 MeV.

(To be continued.)