San José State University
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The Analysis of the Impact
of Additional Neutrons on the
Binding Energy of Nuclides and
the Matter of Magic Numbers

This material examines the relationship between the binding energy of nuclides and additional neutrons. For an illustration consider the Selenium 68 nuclide. It contains 34 neutrons and 34 protons, exactly enough to form 17 alpha particles. Its binding energy is 576.4 million electron volts (MeV). For the selenium nuclide with 34 protons and 35 neutrons the binding energy is 586.62 MeV, an increase of 10.72 MeV over Selenium 68 nuclide. If a second neutron is added the binding energy goes up by another 13.71 MeV, a larger increase than for the first neutron. The total increase for the two neutrons over the binding energy for the Se 68 nuclide is 23.93 MeV.

Altogether 24 neutrons can be added to the Se 68 nuclide and still have a nuclide stable enough to have its mass measured. The data for the increase over the binding energy of the Se 68 are shown below.

This appears to be a remarkably smooth curve, possibly a parabola. However there are subtle characteristics which show up in terms of the increments of binding energy as additional neutrons are added; i.e.,

The alternation of smaller increases followed by larger ones is undoubtably due to the formation of spin-pairs of neutrons. To get the general shape of the relationship between increases in binding energy and additional neutrons it is necessary to smooth out the alternating fluctuations. This can be done in a number of different ways, but the method used here is to take a simple average of the alternating increases. The data for these smoothed increments are as follows.

For a parabola the increments would follow a straight line as a function of the the number of neutrons. The curve above is roughly a straight line with a drop at about 16 neutrons.

For a parabola the increments in the increments would be constant. This is not quite the case but there is a definite tendency toward that condition except for the drop around the level of 16 added neutrons.

The fluctuations about the smoothed curve are of interest.

This plot shows that the amplitude of the fluctuations is not constant. There is a slight trend and then a substantial drop at the same number of neutrons. This shows up a bit better if the absolute values of the deviation from the smoothed values are plotted, as below.

Although the above approach is reasonable and has its virtues there is another approach that is conceptually sounder. When a neutron is added which does not get paired its effect on the binding energy is the pure effect of the another neutron. When a neutron is added which does get paired its effect is the sum of the effect of adding a neutron and the enhancement of binding energy due to a pair formation. Here is the plot of data for the first, third and other odd numbered additions of neutrons.

The plot is reasonably close to a straight line. The regression equation for this line is

ΔB = 10.37458 − 0.32437n
R² = 0.986

The enhancements in binding energy due to the addition of a neutron which completes a pair are

The average level for the first eight is 3.175 MeV and 1.458 MeV for the last four.

The analysis would be simple if the functional relationships for all alpha nuclides were as simple as those for selenium.

A Survey for the General Form of the Functional Relationships

The case of tin is a good place to start the survey. Tin has more stable isotopes than any other element. The largest alpha nuclide is Sn100. Thirty seven neutrons can be added to the Sn100 nuclide. The increments in binding energy as those neutrons are added are as follows.

Viewing the lower edge points as the pure effect of adding neutrons one sees four things. First, there is the downward slope. Second, the downward slope seems to decrease as more neutrons are added. The downward slope appears to be constant up to the level of about 16 added neutrons and thereafter becomes less negative.

The plot of the increments in the increments is as follows.

Third there is a drop in the level of the increments at about 32 added neutrons. Fourth, the enhancement due to neutron pair formation constant up to the point where the incremental effect drops. There the pairing enhancement drop also. This is shown below.

Although the incremental effect appears to be piecewise linear it can be reasonably approximated by a quadratic function of the number of added neutrons. The enhancement to pairing appears to be constant over intervals of added neutrons. In the case of Sn100 it can be represented by two levels.

The next smaller alpha nuclide is Cd96. It sustains the addition of 34 neutrons. The increments in binding energy as those neutrons are added are as follows.

For the cadmium case the drop in the level comes after about four neutrons rather than 32 as in the case for tin. Again the relationship for the unpaired neutrons (the lower points) appears to be piecewise linear with the downward slope becoming less negative at about 16 neutrons and at 24 neutrons. The enhancement due to pairing seems to be constant over four ranges: 1 to 4 neutrons, 5 to 16 neutrons, 17 to 24 neutrons and 25 to 34 neutrons.

The case of palladium 92 qualitatively duplicates that of cadmium 96. The break points come at the same numbers of neutrons. The levels are different but the shapes are quite similar.

The case for ruthenium 88 provides an interesting variation from the previous two alpha nuclides. Its breakpoint comes after the addition of six neutrons.

There appears to be little or no change in the enhancement of binding energy due to neutron pairing.

The case of the molybdenum nuclides is next. The increments in binding energy for neutrons added to the Mo84 nuclide follow the general pattern found above.

The breakpoint is at eight neutrons.

For the zirconium nuclides there is breakpoint at ten neutrons and another at 17 neutrons. This is a rare instance in which the breakpoint occurs at an odd number of neutrons. Otherwise it is the pattern observed before of piecewise linear declining incremental increases and piecewise constant enhancement for neutron pair formation.

For the strontium nuclides there is an apparent decrease in the enhancement of binding energy due to neutron pair formation and then a subsequent increase.

The graph for the krypton nuclides is quite similar to that for the strontium nuclides.

The graph for the selenium nuclides was shown previously. Here it is again.

Continuing with the results for germanium and zinc.

The next graph is for nickel and it does not display any drop, whereas the following one for iron does.

The pattern continues with chromium.

And with titanium as well.

And with calcium.

Argon displays what appears to be two breakpoints; one after the addtion of 2 neutrons and another after the addition of 10.

Sulfur also displays two breakpoints; one after 4 neutrons and the other after 12.

For silicon there is no second breakpoint.

For magnesium there is some uncertainty as to whether the breakpoint occurs at 2 or 4 neutrons.

With neon there is a definite break after 4 neutrons.

Likewise for oxygen there is definite break after 6 neutrons.

For carbon there is different pattern.

Beryllium fits into the pattern for carbon.

Finally there is helium with a unique profile.

Compilation of the Results

Before the results are presented it is important to note that there is model of nuclear structure called the shell model. In that theory there is a set of numbers, called magic numbers, which represent the capacity of certain structures in nuclei. Those numbers are 2, 8, 20, 28, 50, 82 and 126. For more on this topic see the shell model of nuclear structure.

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

Conclusions So Far

The results are an overwhelming vindication of the magic numbers of 8, 20, 28, 50 and 82 from the shell theory of nuclear structure. On the other hand, if the results vindicate 8, 20, 28, 50 and 82 as magic numbers it means that 6 and 14 are also magic numbers.

The ambiguities concerning nickel and magnesium were resolved to fit the pattern for the other nuclides.

In the case of argon and sulfur there were second breakpoints. At those second breakpoints the number of neutrons is 28, further confirming the shell model.

The fact that these numbers are all even indicates that neutrons are added into some arrangements in nuclides in pairs.

(To be continued.)

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