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The Statistical Explanation of
the Relationship Between Incremental Binding
Energy and the Proton Number of a Nuclide

There exists a marvelously simple and exact relationship between the ionization energy of an electron in an atom or ion and the number of protons in the nucleus. The energy required to separate an electron from an atom or ion is a quadratic function of the net attractive charge experienced by the electron; i.e., the number of protons in the nucleus less the shielding due to electrons in inner shells or in the same shell as the electron. An analogous but more complex relationship may exist between the incremental binding energy of a neutron and the number of protons in the nucleus. The incremental binding energy for a neutron in a nuclide is the binding energy for that nuclide less the binding energy of a nuclide having one less neutron.

This material is an investigation of the form of the relationship between incremental binding energy and the number of protons (proton number) of the nuclide. This investigation starts with the case of the 28th neutron. The data are shown in the graph below.

There is an at least roughly linear relationship between the incremental binding energy (IBE) and the proton number (#p). The regression of IBE on #p gives

IBE = −6.45813+ 0.78772#p
[-11.6] [33.5]
R² = 0.98506

The numbers in the square brackets below the regression coefficients are the t-ratios for the coefficients; i.e., the ratios of the coefficients to their standard deviations. The magnitude of the t-ratio for a coefficient must be at least about 2 for the coefficient to be statistically significantly different from zero at the 95 percent level of confidence. The coefficient of determination (R²) indicates that about 98.5 percent of the variation in the IBE is accounted for by variation in #p.

The statisical performance of the simple linear regression in terms of R² and the magnitudes of the t-ratios is reasonably good. However there appears to be a relatively sharp increase in the IBE for the proton number equal to the neutron number 28. (This jump in IBE when the proton number equals the neutron number occurs for all of the cases previously investigated. See IBEPN1.) Such a jump phenomenon can be included in the regression equation by creating a variable d(z) where d(z)=1 if z≥0 and zero otherwise. Thus the regression equation is

IBE = c0 + c1#p + c2d(#p-28)

The regression using this equation gives

IBE = −4.77752 + 0.69792#p + 1.46252d(#p-28)
[-9.8] [29.8] [5.0]
R² = 0.99420

The inclusion of a jump in IBE at #p=28 improves the statistical performance of the regression and the t-ratio confirms the significance of the jump phenomenon.

In nuclear binding energy relationships there is often a sawtooth pattern that reflects the effect of the formation of spin pairs. There is a hint of such a pattern in the graph of IBE versus #p. This effect can be tested for statistically by creating a variable d(#p) which is 1 if #p is even and 0 if it is odd. The regression resulting from the inclusion of such a variable is

IBE = −5.08296 + 0.70353#p +
1.37113d(#p-28) + 0.38080e(#p)
[-11.8] [35.2]
[5.5 ] [2.7]
R² = 0.99608

The result indicates that having a even number of protons results in an increment of about 0.38 MeV in the incremental binding energy. Statistically this effect is significant but of minor importance.

It is possible that there might not only be a jump in the level of the IBE when #p becomes equal to the neutron number 28 but also a change in the slope of the relationship. This can be tested for by creating a variable u(z) such that u(z)=0 sfor z<0 and u(z)=z for z≥0. The inclusion of such a variable gives the following regression results

IBE = −5.19984 + 0.70913#p
+ 1.58057d(#p-28) + 0.38497e(#p)
−0.13153u(#p-28)
[-12.2] [35.7]
[5.5] [2.8]
[-1.4]
R² = 0.99654

Although there was some small improvement in the coefficient of determination for the inclusion of a change in slope at #p=28 the t-ratio of -1.4 indicates that the slope above #p=28 is not significantly different from the slope below #p=28. Thus for further statistical analysis of the relationship of IBE to #p can assume the slope is unchanged beyond the point at which the proton number equals the neutron number.

Regression equations were computed for #n=27, 29, and 30. The results are tabulated below.

#nConst.#pd(#p-#n)u(#p-#n)e(#p)
27-8.116270.7555111.33281-0.08547-0.416170.99323
28-5.199840.709131.58057-0.131530.384970.99654
29-7.852670.665421.13892-0.05462-0.314820.99653
30-7.471370.699581.300300.024870.170240.99713

The coefficients high-lighted in white are not statistically significant at the 95 percent level of confidence. The statistical insignificance of the coefficients for u(#p-#n) indicates the slope of the relationship above the jump at #p=#n is not different from the slope below that value.


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