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The Relationship Between
the Incremental Binding Energy
and the Number of Additional
Neutrons in a Nuclide

Consider the binding energy for a sequence of isotopes of an element. The incremental binding energy is the difference between the binding energy of isotopes that differ only by one have one more neutron than the other. To avoid the effect of an added neutron being sufficient for the formation of an alpha particle the sequence starts out with a nuclide that could contain an integral number of alpha particles.

The graph of the binding energy for a sequence starting with the selenium 68.

The pattern of the incremental increases in binding energy is more revealing.

The point where the relationship changes for selenium is where 16 neutrons are added to the 34 that are already there. This means that the break in the relationship occurs when there are 50 neutrons in the nuclide. The number 50 is one of so-called magic numbers. In a previous study it was found for all the alpha nuclides the breakpoints come when the neutron total is equal to a magic number (if six and fourteen is added to the set of magic numbers). The compilation of the data is as follows.

The Number of Neutrons in Nuclides at Breakpoints
ElementProtonsNeutronsAdded Neutrons
to Breakpoint
Total Neutrons
Tin Sn10050503282
Cadmium Cd964848250
Palladium Pd924646450
Ruthenium Ru884444650
Molybdenum Mo844242850
Zirconium Zr8040401050
Strontium Sr7638381250
Krypton Kr7236361450
Selenium Se6834341650
Germanium Ge6432321850
Nickel Ni562828028
Iron Fe522626228
Chromium Cr482424428
Titanium Ti442222628
Calcium Ca402020828
Argon Ar361818220
Sulfur S321616420
Silicon Si281414620
Magnesium Mg241212214
Neon Ne201010414
Oxygen O1688614
Carbon C126628
Beryllium Be84426
Helium He42246

Consider the set of alpha nuclides for which the breakpoint comes at a neutron number of 50. These elements are cadmium, palladium, ruthenium, molybdenum, zirconium, strontium, krypton, selenium, germanium and zinc. The element tin can also be construed as belonging to this set. It might be expected that the incremental energy for the cases beyond the breakpoint are the same. Here is the graph for tin, cadmium, palladium, ruthenium and molybdenum.

To a remarkable degree the relationships of binding energy and neutrons beyond the breakpoint have the same slope and the same amplitude of the fluctuations. This suggests that an appropriate empirical relationship would be of the form

ΔB = c0 + cP + c2n + c3u

where ΔB is the incremental binding energy, P is proton number, n is the number of neutrona beyond the breakpoint, and u=0 if n is odd and u=1 if n is even. This last term is the binding energy enhancement for the formation of a neutron pair.

For elements associated with a breakpoint at 50 neutrons this regression equation gives the results:

ΔB = -10.0146 + 0.41687P -0.20549n + 2.28107u
  [-19.7]        [51.2]        [-39.2]        [30.9]
R² = 0.95234

The coefficient of determination, R², for this equation was 0.95234. This means the correlation between the binding energy increments and the values predicted by the equation is 0.976. Another measure of the goodness of fit is the standard deviations of the unexplained variation. For the above equation this standard error of the estimate is 0.509 MeV. The numbers in brackets below the coeficients are the t-ratios (ratio of the coefficient to its standard error).

The numerical result which is quite notable is the value for enhancement of binding energy due to a neutron pair formation, 2.28107. This is quite close to the binding energy for the formation of a proton-neutron pair, a deuteron, 2.22457 MeV. The difference is 0.0565 MeV and the standard deviation of the estimate of the coefficient is 0.0738 MeV. Thus the estimated enhancement in binding energy is not significantly different from the binding energy of the deuteron at the 95 percent level of confidence.

There appears to be a slight upward curvature to the relationships between incremental binding energy and the number of added neutorns. The statistical fit might be improved by including a quadratic term for the neutron number. The results for such a regression are:

ΔB = -9.56717 + 0.41671P -0.3012n + 0.0003347n² + 2.29567u
  [-21.1]        [6.3]        [-21.0]        [7.1]        [34.9]
R² = 0.96244

There is a slight, 1%, increase in the coefficent of determination from taking into account the curvature of the relationship, but this not the major source unexplained variation. The standard error of the estimate was improved from 0.509 MeV to 0.453 MeV. A regression equation that allows for variation in the slope as a function of the proton number resulted in only a very small improvement in the coefficient of variation.

The really significant difference between the linear and the quadratic equation is the value of the linear coefficient, the coefficient of n. For the linear equation that coefficient is −0.20549 whereas in the quadratic equation the coefficient of n is −0.3012. Since the coefficient of n contains important information concerning the structure of the shells it is important to use the quadratic regression for further analysis even though there is not much improvement in the statistical fit compared with the linear equation.

The elements which are associated with a breakpoint of 28 neutrons are nickel, iron, chromium, titanium and calcium. The regression results for this set of elements are:

ΔB = -9.1401 + 0.691957P -0.37572n + 2.432633u
  [-18.0]        [28.7]        [-29.1]        [19.9]
R² = 0.95912

The results indicate a sharper downward slope of the relationship for the case of the 28 neutron breakpoint elements compared to those for the 50 neutron breakpoint. Also the enhancement for pair formation was larger for the 28 neutron set compared to the 50 neutron set. There may also be a dependence of the slope of the relationship on the proton number. The elements argon and sulfur have a secondary breakpoint for 28 neutrons and their values could have been included in the regression but were not.

(To be continued.)

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