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The Statistical Explanation of
the Relationship Between Incremental Binding
Energy and the Proton Number for all Nuclides

This is an investigation of the statistical relationship between the incremental binding energy of a neutron and the number of protons in the nuclide. 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. Binding energies are available for 2931 nuclides but the incremental binding energy (IBE) can be computed for only 2816 nuclides.

The data are shown in the graph for the case of the 28th neutron.

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 e(#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 at the 95 percent level of confidence from the slope below #p=28. Thus for further statistical analysis of the relationship of IBE to #p one should consider the possibility that the slope is unchanged beyond the point at which the proton number equals the neutron number.

Regression equations were also 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.

The Statistical Results for all 2816 Nuclides
for which the Incremental Binding
Energy of a Neutron is Defined

The statistical results in the previous table indicate that the effects depend upon #n, the neutron number. To allow for a variety of dependences the relationship is allowed to be quadratic; i.e., that it depends upon #n*#n, #p*#p and #n*#p as well as #n and #p. To capture the jump in IBE for #p equally or exceding #n variable d(#p-#n) is included which is 0 if #p<#n and 1 otherwise. Likewise a variable u(#p-#n) is included to allow for the possibility that the slope of the dependence of IBE on #p could change for #p≥#n. Furthermore these effects might depend upon #n so two additional variables #n*d(#p-#n) and #n*u(#p-#n) are included in the regression. The results are as follows.

VariableCoefficientt-Ratio
#n -0.512618308 -43.5
#p 0.805305664 45.0
#n*#n -0.00130124 -3.9
#n*#p 0.008153716 8.1
#p*#p -0.009468075 -12.0
d(#p-#n) 8.26645333520.9
u(#p-#n) 0.8300048244.8
#n*d -0.163532574 -10.6
#n*u -0.033278151 -4.1

The coefficient of determination (R²) for this regression is 0.80732.

The negative value for the coefficient for #n and the positive value for the coefficient for #p is further evidence that neutrons repel each other whereas proton and neutrons attrack each other.

The value of 8.266453335 MeV for the jump in the incremental binding energy when the proton number equals the neutron number indicates that this effect is quantitatively significant and its t-ratio of 20.9 indicates that it is highly significant statistically. The regression coefficient for #n*d(#p-#n) of -0.163532574 indicates that the magnitude of the jump is smaller for higher values of #n. At about #n=50 the jump becomes insignificant and above 50 becomes a drop instead of a jump. However at higher values of the neutron number there are no nuclides which have a proton number that high. The highest value of #n such that there is a nuclide with #p=#n is #n=46. The highest value of #n such that #p goes beyond #n is #n=37.

The surprise is that the coefficient for u(#p-#n), representing a change in the slope of the relationship between IBE and #p at #p=#n is statistically significant at the 95 percent level of confidence. The effect of #n on this change in slope is also statistically significant. The value for #n*u of -0.033278151 just about cancels out the value of 0.830004824 for u(#p-#n) at #n=25. So the lack of a significant change in the slope at #p=#n for #n in the range of 27 to 30 was just a coincidence having to do with that particular range of values for #n.

When variables for an even-odd relationship were included they were found not be statistically significant at the 95 percent level of confidence.


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