San José State University
Thayer Watkins
Silicon Valley
& Tornado Alley

Neutrons repel each other through the strong force,
as do protons. Nuclei are held together by the
attractions of neutrons and protons.

It is well known that too many protons relative to the number of neutrons in a nuclide results in instability. This is hypothetically explained by the mutual electrostatic repulsion of protons beyond a certain distance. It is hypothesized that there is a strong force attraction between protons but this attraction drops off rapidly with separation distance. Thus for larger nuclei some of the protons are necessarily at distances from each other where the electrostatic repulsion overwhelms the strong force attraction. Therefore for stability a nuclide needs an excess of neutrons to offset the electrostatic repulsion of the protons.l

This is all perfectly plausible but what accounts for the fact that too many neutrons compared to the number of protons also results in instability? This is one element of the evidence that neutrons repel each other.

The Nucleon Hypothesis

When the neutron was discovered in early 1930's Werner Heisenberg conjectured that the proton and the neutron were essentially the same particle; the proton just had a positive charge turned on. This supposed common particle was called the nucleon. Nucleons were then supposed to experience a strong force attraction for each other. This hypothesis was generally accepted among physicists.

The picture is complicated by the fact that neutrons and protons form spin pairs. This makes it appear that there is an attractive force between two nucleons. But spin pair formation is exclusive in the sense that one neutron can form a spin pair with only one other neutron and with one proton. This force holds spin pairs together but there must be another force holding nuclei together. This is the true nuclear strong force and it shows up only in the properties of larger aggregates of nucleons. The interaction of a small number of nucleons, as in scattering experiments, is dominated by spin pair formation. The stability of larger nuclei is determined by the non-exclusive strong force interactions of nucleons.

The Quark Model

In the early 1960's Murray Gell-Mann and George Zweig independently developed the theory that elementary particles are composed of still more fundamental entities, subsequently called quarks. Both were at the California Institute of Technology at the time. According to the quark theory protons and neutrons are made up of three quarks each. A proton is composed of two up quarks and one down quark; whereas a neutron is composed of one up quark and two down quarks. Thus the proton and neutron are not really the same particle. The interactions of protons and neutrons in various combinations is the result of the interaction of their constituent quarks.

The Pauli Exclusion Principle

One rationale for the mutual repulsion of particles of the same type is that, according to the Pauli Exclusion Principle, two particles of the same type cannot have the same set of quantum characteristics in the same space.

Nucleonic Balance

Nuclear stability is represented in terms of binding energy. If the binding energy of all nuclides having the same number of nucleons is plotted versus the number of protons the result is a curve that rises to a peak where the numbers of protons and neutrons are approximately equal. This is shown below for 24 nucleons.

The maximum binding energy occurs for Magnesium 24 with 12 protons and 12 neutrons. It is very significant that the binding energy drops off in both directions.

This suggests that the binding energy (BE) of nuclides might be explained in terms of the number of interaction pairs of nucleons. If the numbers of protons and neutrons are denoted by #p and #n, respectively, the number of proton-proton interaction pairs (#pp) is equal to ½#p(#p-1) and likewise the number of neutron-neutron interaction pairs (#nn) is equal to ½#n(#n-1). On the other hand the number of proton-neutron interaction pairs is #p*#n.

There is another type of pairing of nucleons, spin pairing. Spin pairing, in contrast to the interaction pairing, is exclusive. That is to say, if a proton forms a spin pair with another proton then it cannot form a spin pair with any other proton. The number of proton spin pairs is (#p%2), the number of protons divided by 2 and the result rounded down to an integer and likewise for the number of spin pairs of neutrons is (#n%2). The number of possible spin pairs of protons and neutrons is the minimum of #p and #n, min(#p, #n).

The regression of the binding energies of the 2931 nuclides gives:

BE = -0.48958#pp + 0.27804n*p − 0.19254#nn
+ 4.310005984(p%2) + 10.31174min(p,n)
+ 13.83460453 (n%2)
[-40.8] [35.9] [-37.3]
[8.3] [34.7]
R² = 0.999882068

The numbers shown in the brackets are the t-ratios for the coefficients, the ratio of the coefficient to its standard deviation. For a coefficient to be statistically significantly different from zero at the 95 percent level of confidence its magnitude must be at least about 2. The t-ratios are all very much greater than 2 in magnitude.

What the results mean is that there is a mutual repulsion between protons and between neutrons, although to a lesser extent. All three types of spin pairing have a positive effect on binding energy but that for proton-neutron pairing is much stronger than for proton-proton spin pairing.

It is worthwhile to examine the quantities for the case of Magnesium 24 where #p=12 and #n=12. The numbers of proton-proton interaction pairs #pp and neutron-neutron interaction pairs #nn are both 66. The number of proton-neutron interaction pairs is 144. The numbers of proton-proton spin pairs and neutron-neutron spin pairs are both 6. The number of proton-neutron spin pairs is also 6.

When these quantities are entered into the regression equation the result is:

BE = -32.31211648 + 40.03743633 -12.7079297
25.8600359 + 123.7408633 + 83.00762717

What the figures indicate is that the major part of the binding energy arises from the spin pairing of the nucleons, particularly the proton-neutron spin pairing. However the spin pairs are substructures and they have to be held together by interaction pairing. The interaction pairing is dominated by the proton-neutron pairing but in this case overall the interaction pairing nets out to a small negative quantity. This represents the inaccuracies of the regression equation. The actual binding energy of the Mg 24 nuclide is 198.25689 MeV whereas the regression equation estimate is 227.62591652 MeV.

The Shell Occupancy Model with Same Shell Shielding

The protons and neutrons in a nucleus are arranged in shells, perhaps of some concentricity. Likewise the electrons in an atom or ion are arranged in concentric shells. Those electrons are mutually repelling. The electrons in inner shells shield the electrons in outer shells from some of the positive charge of the nucleus. The ones in the same same shell also shield some of the positive charge but not fully. As electrons are added to a shell the energy required to remove an electron decreases due to this shielding by electrons in the same shell. If the electrons were mutually attracting the effect of the electrons in the same shell would be to enhance the ionization energy.

In the nucleus when neutrons are added to a shell the incremental binding energy decreases except for the phenomena of pair formation. This indicates that overall neutrons are mutually repelling. The protons are arranged in shells as well and the decline in incremental binding energy with increased occupancy of the shell confirms what is already known; i.e., that protons are overall mutually repelling. The incremental binding energy declines more sharply with the increased occupancy of the proton shell than does that of the neutrons. Thus neutrons are mutually repelling but not as strongly so as protons.

The Incremental Binding Energies

The incremental binding energy for a neutron in a particular nuclide is the difference between the binding energy of that nuclide and that of a nuclide that has one less neutron. For example, the binding energy of the Mg 24 nuclide with its 12 protons and 12 neutrons is 16.53209 MeV. The incremental binding energy of the nuclide with 13 protons and 12 neutrons (Al 25) is 16.9322 MeV. It is very reasonable that the binding energy should be larger for the nuclide with one more proton for the neutron to interact with.

Now consider the nuclide with one more neutron instead of one more proton; i.e., the nuclide with 12 protons and 13 neutrons (Mg 25). Its binding energy is 7.33067 MeV. The figures for the incremental binding energy are afffected by the formation or nonformation of neutron spin pairs. The incremental binding energy of the nuclide with 12 protons and 14 neutrons (Mg 26) is 11.09307 MeV. In this case a neutron was two more neutrons to interact with and still the incremental binding energy is less. This is because there is a repulsion against the neutron due to those two extra neutrons. The ratio of the per neutron decrease to the increase for another proton is 0.5(16.53209-11.09307)/(16.9322-16.53209)=2.71951/0.40011=6.79690585.

For another example, consider the nuclide with 25 protons and 25 neutrons (Mn 50). Its incremental binding energy is 13.0818 MeV. An increase of one proton results in an incremental binding energy of 13.814 MeV. On the other hand, a nuclide with two more neutrons (Mn 52) has a binding energy of 10.5355 MeV. This is decrease in incremental binding energy of about 1.25 MeV per additional neutron compared with an increase of 0.73 MeV for one proton. The ratio is 1.7.

As a third example, take the nuclide with 60 protons and 90 neutrons. It is Nd 150 and its incremental binding energy is 7.38 MeV. An additional proton raises the incremental binding energy to 7.863 MeV. On the other hand two more neutrons lowers IBE to 7.275 MeV. In this case the ratio of the per neutron decrease to the increase for the proton is only about 0.11, but the important fact is that incremental binding energy decreases for addition neutrons whereas it increases for an additional proton. A demonstration that this pattern prevails for all nuclides would confirm that between neutrons there is a repulsion instead of an attraction.

A first step in that direction is to examine an expanded version of the last example. Here are the data for all the nuclides having 90 neutrons. The changes in incremental binding energy are for two more protons and two more neutrons to avoid conflating the effects of pair formation with the effect being examined.

NuclideProtonsNeutronsChange in IBE
Due for Two
More Protons
Change in IBE
Due for Two
More Neutrons
144Xe 54 90 0.800 -0.300
145Cs 55 90 0.540 -0.310
146Ba 56 90 0.110 -0.480
147La 57 90 1.160 -0.030
148Ce 58 90 1.210 -0.050
149Pr 59 90 0.476 0.354
150Nd 60 90 1.060 -0.105
151Pm 61 90 1.287 -0.376
152Sm 62 90 0.878 -0.291
153Eu 63 90 0.687 -0.398
154Gd 64 90 0.636 -0.358
155Tb 65 90 0.621 -0.426
156Dy 66 90 0.547 -0.386
157Ho 67 90 0.319 -0.268
158Er 68 90 0.529 -0.411
159Tm 69 90 0.620 -0.470
160Yb 70 90 0.520 -0.460
161Lu 71 90 0.270 -0.430
162Hf 72 90 0.495 -0.445
163Ta 73 90 0.330 -0.040
164W 74 90 0.425 -0.246
165Re 75 90 0.370 0.010
166Os 76 90 0.360 -0.180
167Ir 77 90 0.680 -0.360

This data plotted in a scatter diagram is as follows:

The changes in IBE due the increase in protons are all positive and the changes in IBE due the increase in neutrons, except for two, are all negative. One of those nonnegative changes is essentially zero. The other is definitely positive and is an anomally. Nothing is special about the proton number of 59 for that case. It is not a nuclear magic number or such. For now it is simply a peculiar exception. The rest of the cases confirm the assertion that the force between neutrons is a repulsion.

One More Observation

Consider the stability of a shell consisting of particles which attract each other. There is the example of the solar disk consisting of masses subject to mutual gravitational attraction. That system evolved into one in which each shell consisted of a single planetary system, a planet and its satellites. That is likely what would happen in a neutron shell if the neutrons attracted each other. Since that apparently does not happen in the neutron shells, some of which consist of as many as 44 neutrons this is one more argument that neutrons do not attract each other under the nuclear strong force but instead repel each other.

For the corresponding analyis for protons see Proton Repulsion.


There is much left to be done concerning this matter, but the evidence is clear that while the strong force between protons and neutrons is an attraction it is a repulsion between neutrons. This should not be too much of a surprise; it is just another case of like particles repelling each other.

HOME PAGE OF applet-magic
HOME PAGE OF Thayer Watkins