San José State University
Thayer Watkins
Silicon Valley
& Tornado Alley

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 Se68 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.

The relationship changes its form when 16 neutrons are added to the 34 already in the Se 68 nuclide to give 50 neutrons. It is as though some structure has been completed when the number of neutrons reaches 50 and thereafter another structure is is being formed. The number fifty is one of the magic numbers of nuclear stability. The sequence of magic numbers is 2, 8, 20, 28, 50, 82 and 126. It was found in a previous study that the breakpoints in the relationships between binding energy and the number of neutrons added to nuclides that could contain an integral number of alpha particles come when the number of neutron reaches a magic number, if the numbers 14 and 6 are included among the magic numbers. For example, the tin 100 nuclide contains 50 protons and 50 neutrons. Thirty seven neutrons can be added to this nuclide. After 32 neutrons are added the relationship changes form, as shown.

Fifty plus 32 is 82, a magic number.

For the Oxygen 16 nuclide the relationship is

The breakpoint comes when six neutrons are added to the eight already there. Eight and six are 14 a number not traditionally considered a magic number.

Here is the compilation of the results for all of the alpha nuclides, the nuclides which could contain an integral number of alpha particles.

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

Argon and Sulfur have two breakpoints in the relationships.

At the first breakpoints the number of neutrons is 20, a magic number, and at the second the number of neutrons is 28, another magic number. The results confirm the special character of the traditional magic numbers but also indicate that 14 and 6 should be included among the magic numbers.

The previous results were based upon adding neutrons to the alpha nuclides. What is consider here is the same type of analysis applied to the nuclides which contain an integral number of alpha particles plus a proton-neutron pair. These will be called the alpha-plus nuclides. For example, Li6 is an alpha-plus nuclide because it contains 3 protons and 3 neutrons, enough for one alpha particle plus a proton and neutron for a pair.

The results for the alpha-plus nuclides confirm the results for the alpha nuclides, but there are some additional results as well. Consider the relationship for the scandium 42 nuclide.

The breakpoint for scandium occurs for 7 added neutrons, which with the 21 neutrons in Sc 42 brings the total to 28, a magic number.

In the case of the potassium 38 nuclide there are two breakpoints; one after the addition of one neutron and the other after the addition of 9 neutrons.

Thus the breaks in the relationship occur when there are 20 neutrons and 28 neutrons, both magic numbers.

There is an extraordinary difference between the case for the alpha-plus nuclides and the alpha nuclides. For the alpha nuclides as neutrons are added the pattern is a smaller increase then a larger increase. This was attributed to the formation of neutron pairs. For the alpha-plus nuclides the larger increase comes when the first neutron is added. The alpha-plus nuclides have a neutron paired with a proton yet when another neutron is added apparently a neutron pair is formed. What this means that there are two independent pairing processes involved; one for neutrons with protons and another with neutrons with neutrons. The results also indicate that the extra proton does not count in the process of achieving a magic number count. Thus the magicalness is in terms of the neutrons per se and not in terms of the number of nucleons (protons and neutrons).

For most cases the breakpoint or points are unambiguous. For some however it is difficult to discern a clearcut break. For example, consider the case of manganese.

Manganese 50 has 25 neutrons so the addition of 3 neutrons would bring the number of neutrons to the magic number of 28. In the graph a breakpoint at 3 added neutrons is plausible and there is no contrary evidence but it would not have been chosen from simple observation of the relationship.

There are also some cases for which there appears to be something in the nature of a breakpoint which does not correspond to a magic number of neutrons. Consider the case of Cobalt.

There is a breakpoint at one neutron, as would be expected for the Co54 nuclide having 27 neutrons. There appears also to be somesort of breakpoint at 11 neutrons rather than 13. This makes the number of neutrons at the second breakpoint 48 rather than 50.

A Survey for the General Form of the Functional Relationships

The most striking graphics occur for the larger nuclides. The largest alpha-plus nuclide is Indium 98.

There are two breakpoints in the relation. The first occurs when one neutron is added so the total neutron count is 50. The second occurs where 33 neutrons are added and the neutron count is 82. Both 50 and 82 are magic numbers.

The peaks in the incremental binding energy come with the addition of one neutron and the apparent formation of a neutron pair with the neutron that is already paired with the additional proton. Another notable feature of the relationship is that the enhancement in the binding energy due to the neutron pair formation is generally constant of the interval from 1 to 33 neutrons but there is a tapering off of the value towards the end of the interval.

The relationship for the Siver nuclides is quite similar to that for the Indium nuclides except that the maximum number of neutrons to be added does not reach the 35 level required to show a breakpoint at 82 neutrons. The breakpoint at 3 neutrons corresponds to 47=3=50 neutrons.

For the Rhodium nuclides the pattern is same and quite clearly the breakpoint comes at the addition of 5 neutrons which brings the total neutrons to 50.

The pattern continues with the rare Technetium nuclides. There is clearly a breakpoint at the addition of 7 neutrons, which brings the total number of neutrons to 50.

The relation for Niobium nuclides confirms the breakpoint at 50, 41+9, but there is something interesting occurring in terms of the enhancement in binding energy due to neutron pair formation.

As with the Niobium nuclides, the relationship of Yttrium nuclides confirms the breakpoint at 50 neutrons (39+11) but something unusual is happening with the enhancements for neutron pair formation.

For the Rubidium nuclides the peculiarities concerning the enhancement due to neutron pair formation may or may not be there but the confirmation of the breakpoint at 50 (37+13) remains.

With the Bromine nuclides the breakpoint for the addition of 15 neutrons to the 35 already in Br 70 confirms the 50 neutron rule. The peculiarities concerning the enhancement are not there because not enough neutrons can be added to reach the level where they occur.

This is also the case with Arsenic nuclides. However the breakpoint comes with the addition of 17 neutrons which brings the total to 33+17=50.

As mentioned previously, the breakpoint for the Gallium nuclides is not obvious. For confirmation of the 50 neutron total the breakpoint should be 19 additional neutrons. There is no evidence contrary to this value and it is at 19 neutrons that the largest drop occurs.

For Copper nuclides, as with the Gallium nuclides, the breakpoint comes at the end of the sequence of additions of neutrons. The value of 21 additonal neutrons combined with the 29 neutrons of Cu 58 gives a total of 50 neutrons at the breakpoint.

For Cobalt 54 the number of neutrons is 27 and with addition of one neutron the total is the magic number of 28. This is confirmed in the display although it is not obvious that the breakpoint comes at one neutron. There is also something more complex occurring with the level of the enhancement due to neutron pair formation.

With the pattern established it is now clear that for the Manganese nuclides the breakpoint comes with the addition of three neutrons and that brings the total neutrons at the breakpoint to the magic number of 28.

The case for the Vanadium nuclides where the breakpoint is clearly at five neutrons not only confirms the breakpoint at 28 total neutrons for its self but also confirms the preceding case for Manganese.

The case for the Scandium nuclides is of special interest. The breakpoint at seven additional neutrons confirms the break coming at 28 total (21+7) neutrons. However the relationship shows that for the addition of 13 neutrons the enhancement due to a neutron pair formation is missed.

For the Potassium nuclides there is no missed enhancement but there is the interesting phenomenon of two clear breakpoints. One is after the addition of one neutron, which brings the total neutrons to 20, and the second is after the addition of nine neutrons and thus a total neutron level of 28.

For Chlorine nuclides there are also two clear breakpoints as well corresponding to 20 and 28 neutrons. They occur at 3 and 11 added neutrons.

For the Phosphorus nuclides the breakpoints occur at 5 and 13 added neutrons or 20 and 28 total neutrons.

There is an interesting irregularity to the relationship for Phosphorus.

For the Aluminum nuclides the irregularity disappears. The breakpoints are not obvious but knowing they should be at 1 and 7 added neutrons it is easy to see that the breakpoints are indeed at 1 and 7.

For the Sodium nuclides the pattern for the other nuclides suggests there should be a breakpoint at 3 and at 9 added neutrons for total neutron figures of 14 and 20. While the graph does clearly establish these as the breakpoints there is no evidence contrary to those values being the breakpoints.

For the Fluorine nuclides the breakpoint is clearly at 5 added neutrons and thus the total number of neutrons is 14.

For the Nitrogen nuclides are clearly at 1 and 7 added neutrons, corresponding to total neutrons of 8 and 14.

For the Boron nuclides the breakpoints are at 1 and 3 added neutrons, corresponding to total neutrons of 6 and 8.

Finally there is the case of the Lithium nuclides. In this case the breakpoint is at 3 added neutrons and hence 6 total neutrons.

Compilation of Results

The Number of Neutrons in Nuclides
at Breakpoints for Alpha-Plus* Nuclides
alpha-plus NuclideNumber
of Protons
of Neutrons
at First
of Neutrons
at Second
if Present
Indium In98495082
Silver Ag944750 
Rhodium Rh904550 
Technetium Tc864350 
Niobium Nb824150 
Yttrium Y783950 
Rubidium Rb743750 
Bromine Br703550 
Arsenic As663350 
Gallium Ga623150 
Copper Cu582950 
Cobalt Co542728 
Manganese Mn502528 
Vanadium V462328 
Scandium Sc422128 
Potassium K38192028
Chlorine Cl34172028
Phosphorus P30152028
Aluminum Al26131420
Sodium Na22111420
Fluorine F18914 
Nitrogen N147814
Boron B10568
Lithium Li636 
* A nuclide which contains an intregal number of alpha particles plus a proton-neutron pair


There are substructures in nuclides that allow definite maximum numbers of neutrons to be included. These substructures are filled at totals of 6, 8, 14, 20, 28, 50 and 82. (The magic number 126 will be dealt with elsewhere.) These numbers are the traditional magic numbers augmented with the numbers 6 and 14, which undoubtably should be considered as much magic numbers as the traditional ones. The pairing of neutrons is independent of the pairing of neutrons and protons.

(To be continued.)

HOME PAGE OF applet-magic
HOME PAGE OF Thayer Watkins