San José State University
Thayer Watkins
Silicon Valley
& Tornado Alley

Models of Neutron and Proton Shells and Their
Statistical Performance Explaining Binding Energies

There is overwhelming evidence that the protons and neutrons within a nucleus are organized into shells. Furthermore within the shells the neutrons are paired whenever possible and likewise for the protons. There is also binding energy resulting from the fitting of the extra protons and neutrons, if any, into the arrangement of pairs.

The material following reports on the statistical performance of three minimalist models of nuclear shell structure in explaining the binding energies of nuclides.

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. The sum of the squared deviations from the average for a variable is called its 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 Variation is 254,825.2 (MeV)².

Previous work reported on an alpha particle model for explaining binding energies. 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 can be called excess binding energy, is the 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 for the total variation in nuclide binding energies, it is revealed that 92.4 percent of the variation in the binding energies of the 2931 nuclides 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.

The figures of 25 and 41 alpha particles correspond to 50 and 82 neutrons and 50 and 82 protons. The numbers 50 and 82 are so-called magic numbers of nucleons. They correspond to filled shells. For more on nuclear magic numbers and their explanation see explanation of nuclear magic numbers.

Incremental Binding Energies

When additional neutrons are added to a nuclide the binding energy increasings and the amount that it increases depends upon whether the additional neutron forms a pair with a neutron already in the nuclide. This is shown for the element lead (Pb) in the graph below.

The graph shows the breakpoint in the relationship occuring at the magic number level of 126.

In the following let #nn denote the number of neutron pairs in a nuclide and likewise for #pp for proton pairs. Let %n denote the remainder when the number of neutrons is divided by 2. This is 1 if there is an unpaired neutron. Similarly %p denotes the presence of an upaired proton.

The simplest version of a model explaining the binding energies in terms of the numbers of neutrons and protons that takes into account pairing is represented by the equation

BE = c1#nn + c2#pp + c3%n + c4%p

The results of the indicated regression are, in tabular form,


The t-ratio for a coefficient is the ratio of the value of the coefficient to its standard deviation. A t-ratio has to be on the order of 2 or greater to be statistically significantly different from zero at the 95 percent level of confidence. For the above regression the effect of a singleton proton is not significantly different from zero.

The coefficient of determination (R²) for the equation is 0.95935. This means that about 96 percent of the variation in binding energies of the 2931 nuclides is explained by the number of neutron and proton pairs and the existence or not of a singleton neutron or singleton proton. This is not bad in terms of statistical performance but the alpha particle model explained 98 percent of the variation.

A better measure of the statistical performance is the standard error of the estimate, which for the above regression is 101.86 MeV. Since the average binding energy of a nuclide is 1072 MeV this means roughly that the the regression equation gives the binding energies of the nuclides ±10 percent.

It was noted above that there is a bendpoint in the binding energy relationship at 25 alpha particles which corresponds to 25 neutron pairs and 25 proton pairs. When variables are included in the regression that represent the number of neutron pairs and proton pairs above 25 the statistical performance improves dramatically. Such a variable for the neutrons pairs is constructed as follows

u(#nn-25) = 0 if #nn<25
            = #nn-25 otherwise

The results are:


The coefficient of determination for this equation is 0.998342 and the standard error of the estimate is 20.5787 MeV. This means that the equation can give the binding energies accurate to about ±2 percent.

The coefficient for u(#nn-25) indicates that the slope of the relationship increases by 2.90461 MeV per neutron pair at #nn=25. The slope of the relationship between binding energies and the number of proton pairs decreases by 12.7694 MeV at #pp=25.

When bendpoints are allowed for at 41 pairs the coefficient of determination increases ever so slightly from 0.998342 to 0.999023. The standard error of the estimate however is reduced from 20.5787 MeV to 15.8014 MeV, a notable improvement; the regression equation gives the binding energies to ±1.5 percent instead of ±2 percent.


All of the coefficients are statistically highly significant. The negative values for additional protons for larger nuclides is consistent with the theory that at larger distances the electrostatic repulsion between protons becomes dominant over the nuclear strong force attraction.

The Inclusion of Quadratic Terms

The preceding regressions were all piecewise linear. This would mean that the incremental binding energy within a shell is constant. All of the graphs of the relationship between the incremental binding energy and the number of neutrons show a decreasing incremental value as the number of neutron increase. The same applies for increasing numbers of protons. To capture this phenomenon the regression equation must include quadratic terms; i.e., the squares of the number of neutron or protons in a shell.

When these quadratic terms are included in the regression the coefficient of determination improves slightly, from 0.999023 to 0.999211. The standard error of the estimate is reduced from 15.8014 MeV to 14.21778 MeV. The regression coefficients are generally but not all statistically significant. Here are the values.


The Inclusion of More Shells

The graph previously shown indicated that there may be a bendpoints for 2 and 14 alpha particles. These correspond to 2 and 14 neutron pairs and for 2 and 14 proton pairs.

Here are the results of the inclusion of such bendpoints.


The coefficient of determination for this regression is 0.999467 and the standard error of the estimate is 11.7 MeV. Some of the coefficients are not statistically significant at the 95 percent level of confidence.


A simple shell model for nuclides explain 99.9 percent of the variation in the binding energies. The model condenses most of the information for the 2931 nuclides to eight parameters of a regression equation which gives the binding energies of nuclides to an average accuracy of about ±16 MeV or roughly ±1.5 percent.

A more complex model explains 99.95 percent of the variation using 22 parameters. The average accuracy of this model is ±12 MeV or roughly ±1.1 percent.

(To be continued.)

HOME PAGE OF applet-magic
HOME PAGE OF Thayer Watkins