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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 neutron; one having 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 of 50 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 + 0.003347n² + 2.29567u
  [-21.1]        [6.3]        [-21.0]        [7.1]        [34.9]
R² = 0.96244
df=185

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.

Elsewhere there is developed a Shell Occupancy Model that indicates, among other things, that the linear coefficient for the number of additional neutrons should be inversely proportional to the radius of the shell. 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. As in the case of the elements tin, cadmium, palladium, ruthenium and molybdenum, the relationships between incremental binding energies and the number of neutrons beyond the breakpoint of 28 for nickel, iron, chromium, titanium and calcium have similar slopes.

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.

The regression involving a quadratic form dependence on the number of neutrons is

ΔB = -8.09596 + 0.674129P -0.55619n + 0.009215n² + 2.446729u
  [-18.0]        [34.1]         [-17.4]        [6.0]        [24.7]
R² = 0.97355
df=70

Three elements that involve neutrons in the 21-to-28 shell are calcium, argon, sulfur and silicon. The graph of the incremental binding energies for these four elements is:

As is the case for the data shown above for the 29-to-50 and 51-to-82 shells the slopes are similar, as is the enhancement due to pair formation.

The quadratic form regression for the data is

ΔB = -9.99077 + 0.924345P -0.15897n -0.00681n² + 3.324116u
  [-4.5]        [50.2]         [-0.2]        [-0.08]        [4.1]
R² = 0.993445
df=32

The effects of the number of neutrons on the incremental binding energy are not statistically significant.

The picture for the 15-to-20 shell is displayed below using the data for silicon, magnesium and neon.

The quadratic form regression equation is

ΔB = -5.34909 + 1.109505P -1.7588n + 0.14111n² + 3.230869u
  [-12.5]        [18.0]         [-6.1]        [3.5]        [15.3]
R² = 0.98014
df=18

For this shell there is no problem of statistical significance for the coefficients of the regression equation.

If a shell were arbitrarily divided the lower portion would have a steeper slope and the upper portion would have a smaller slope and there would be less influence of the quadratic effect. This suggests the possibility that the 15-to-20 and 21-to-28 shells are not separate shells but instead there is a single 15-to-28 shell. When the data the two supposedly separate shells are combined the quadratic regression equation is

ΔB = -4.2118 + 0.942004P -1.10444n + 0.040327n² + 3.326598u
  [-9.7]        [35.0]         [-15.1]        [9.0]        [26.6]
R² = 0.976326
df=49

The results fit the pattern of the other two shells nicely and this will be taken as confirmation that the two shells are really just parts of one shell.

Summary of Statistical Results

Neutron
Shell
InterceptNumber
of Protons
(Atomic Number)
Number of
Additional
Neutrons
Number of
Additional
Neutrons
Squared
Neutron Pair
Enhancement
Coefficient of
determination
Degrees of
Freedom
15 to 28-4.211180.94200-1.104440.040327 3.3265980.9763349
29 to 50-8.095960.67413-0.556190.009215 2.4467290.9735570
51 to 82-9.567170.41671-0.30120.003347 2.295670.96244185

According to the Shell Occupancy Model developed elsewhere the coefficients for the number of additional neutrons are inversely proportional to the radii of the shells. According to the above results then the radius of the 29-to-50 shell is 1.99 times the radius of the 15-to-28 shell and the radius of the 51-to-82 shell is 1.85 times that of the 29-to-50 shell. According to a quantum theory of two particle systems the radii of particle orbits should be proportional to the square of a quantum number. If the quantum number for the 15-to-28 shell is 2 and those of the next two shells are 3 and 4, respecitively, then the ratios of the radii should be (9/4)=2.25 and (16/9)=1.78.

According to the Shell Occupany Model the ratio of the linear coefficient for additional neutrons to the quadratic coefficient should be proportional to the net charge the shell experienced as it begins to fill. Those ratios are 27.4, 60.4 and 90.0 where as the atomic numbers of the nuclides at which the shells begin to fill are 14, 28, 50. As patterns go in nuclear physics that it not a bad fit; the ratios being 1.96, 2.16 and 1.80, respectively.

(To be continued.)


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