There are a number of different theories about the structure of nuclei, such as the liquid drop model and the shell model. One that has been proposed from time to time is the model that within a nucleus there are substructures of alpha particles, He4 nuclei. This model has not been accepted despite the considerable evidence for its validity. Actually the various models of nuclear structure are not necessarily mutually exclusive. A shell model could accommodate substructures of alpha particles and the shells could be containers of alpha particles. This article examines evidence in terms of the binding energies of nuclides for the existence of substructures of alpha particles.

The most immediate evidence suggesting the existence of alpha particles with nuclei is that some emit alpha particles. This is where the name alpha particle came from. In the early years of the investigation of radioactivity it was found that three types of emissions occurred and they were named alpha, beta and gamma rays. It was subsequently discovered that alpha rays consisted of helium nuclei traveling at high speeds. Beta rays turned out to be electrons traveling high speeds and gamma rays alone were found to be rays, in this case high-powered X-rays.

Binding Energies of Nuclides

The masses of nuclides can be measured by propelling the charged (ionized) nuclei through a magnetic field. The radius of curvature of their trajectories is proportional to their mass and to their speed. The speed is measured by the length of their trajectories in the medium. Thus the masses of all the nuclides have been determined. The mass of a neutron is deduced from the masses of the deuteron and the proton and the dissociation energy of deuterons, the energy of gamma rays required to break apart deuterons.

When the mass of a nuclide is compared with the sum of the masses of the protons and neutrons that make it up it is found that there is generally a mass deficit. (The only exceptions are the H1 nuclide, which is just a proton and thus there can be no mass deficit, and the Be5 nuclide, which has a slight mass surplus and this could be simply the result of an error in the determination of the mass of the neutron.)

The mass deficit is usually expressed as the binding energy of the nuclide, the amount of energy that would be required to break it up. The phenomenon of mass deficits is still an enigma, but it is thought to somehow involve the difference between the potential energy and kinetic energies of an arrangement of nucleons (protons and neutrons) in a nuclide. Thus there is something in the nature of a potential energy well that the nucleus is in. If something reduces the depth of that well then the binding energy is reduced.

Nuclear Force

Protons and electrons are subject to the electrostatic force. A proton and an electron are attacked to each other with a force inversely proportional to the square of their separation distance. Two protons are repelled with a force likewise inversely proportional to the square of their separation distance. This electrostatic force is carried by photons and has an infinite range.

To account for protons being held within a nucleus it is hypothesized that there exists a nuclear force between protons that is stronger than the electrostatic over a short range. Hideki Yukawa theorized that the nuclear force is carried by a particle with a nonzero rest mass. Subsequently pi mesons were discovered and these had the properties required for carries of the nuclear force. These particles decayed and thus the nuclear force had a small effective range. For more on this see Yukawa.

The nuclear force was believed to exist equally between protons and neutrons and between neutrons as well as between protons. This led Werner Heisenberg to speculate that a proton and a neutron are merely different forms of a single particle which he called the nucleon. This nucleon hypothesis came to be widely accepted even though there is considerable evidence against it. It was just a convenient assumption that made theorizing simpler. If Heisenberg's hypothesis were really true there would exist He2 nuclei, two protons bound together by the overwhelming nuclear force. Such a nuclide does not exist. There would also be bound neutron complexes and the emission of gamma rays from neutron collections when such complexes form.

The Integral Alpha Particle Nuclides
and Their Binding Energies

The binding energy of the He4 nuclide, the alpha particle, is 28.3 million electron volts (MeV). This is much greater than that of the He3 nuclide (7.7 MeV), or the H3 nuclide (8.5 MeV). There is apparently some special arrangement of two protons and two neutrons that requires a great deal of energy to break apart. The nuclides having more nucleons than the He4 nuclide have binding energies on the same order as that of the He4 nuclide, indicating that these larger nuclides contain within them a He4 structure.

The nuclides which could contain an integral number of alpha particles are of special interest. When the binding energies of these nuclides are compared, as in the following table, it is found that their binding energies are in excess of what could be accounted for by the formation of alpha particle within them.

As is shown in the table the binding energy of a nuclide which could contain multiple alpha particles is in excess of the binding energies of the alpha particles it might contain. This difference, which will be called the excess binding energy, has the advantage over the straight binding energy that it is not dependent upon the estimated mass of the neutron. Its value depends only upon the measured masses of the charged nuclei. (This is true for all but the alpha particle itself.)

The data in the above table suggests that there are structures of the alpha particles. There is no significant increase in binding energy for two alpha particles but for three there is. The additional binding energy for the number of alpha particles above two is roughly constant at about 7 MeV per additional alpha particle until a level of 14 alpha particles is reached. Thereafter the increase is about 3 MeV per additional alpha particle, as shown below.

The stability of the increments in excess binding energy by computing the increase in binding energy as the number of potential alpha particles increases. This is shown below.

Thus for a nuclide that could contain two to fourteen alpha particles the binding energy increases by the 28.3 MeV of the alpha particle itself plus about 7.3 MeV for the effect of the additional alpha particle on the arrangement within the nucleus for a total of about 35.6 MeV. It is notable that the figure of 7.3 MeV for an additional alpha particle is close to the 7.0 MeV figure for the average binding energy per nucleon in the alpha particle.

For nuclides containing fourteen to twenty five alpha particles the effect on an additional alpha particle on binding energy is an increase of 28.3 MeV for the alpha particle itself plus about 2.70 Mev for the effect of the additional alpha particle on the arrangement within the nucleus. The breakpoint comes at ⁵⁶Ni. It has been long known that there is something special about Iron, Nickel and Cobalt. It is notable that the range over which the increment due to an additional alpha particle is about 7 MeV is from 2 to 14, a span of 12 alpha particles. The range for which the increment is about 3 MeV is from 14 to 25, a span of 11. It appears that there is some arrangement of alpha particles to which an additional one may be added up to a total of twelve. This could be characterized as a shell of alpha particles. Beyond twelve there is a different shell to which alpha particles may be added.

The Evidence that Whenever Possible
Alpha Particles are Formed in Nuclei

Consider the following sequence of nuclides. Starting with an integral alpha particle nuclide a proton is added, then a neutron, then another proton and finally another neutron. The end result is another integral alpha particle nuclide. The binding energies for the first four repetitions of this sequence are shown below.

The black marks indicate integral alpha particle nuclides. The jump in binding energy when another alpha particle can be formed is extremely significant. Up to the level where an alpha particle can be formed the increases in binding energies are small, then when the missing neutron for an alpha particle is added the binding energy jumps upward by the amount of the binding energy of the alpha particle plus an increment for the fitting of that alpha particle into some arrangement of alpha particles. Subtracted from that increase is the small potatoes increase that occurred when the first three nucleons were added.

For larger nuclides the picture is somewhat modified. Below is shown the data for the repetitions of the sequence for the largest nuclides. In the graph the binding energies shown are those in excess of 640 MeV. The 640 MeV figure is just an arbitrary value to bring the level of the values down to the origin of the graph.

Here the increase in binding energy when there is a proton/neutron pair is more significant. It is of a magnitude on the order of half of the increase due to the formation of an alpha particle; i.e., about 14 MeV compared to 30 MeV. Thus the change in the binding energy when an alpha particle can be formed is about (30-14)=16 MeV. The figure of 14 MeV for the effect of a proton/neutron pair is notable for being the binding energy per neutron in the alpha particle.

Now that the nature of the data has been introduced it is appropriate to view the result for the entire sequence from 0 up to 23 alpha particles (⁹²Pd).

In the above graph all of the higher peaks correspond to the formation of integral alpha particle nuclides. The secondary peaks correspond to nuclides which are integral alpha particles plus a proton/neutron pair. These secondary peaks become larger for the larger nuclides, indicating that there is a place for proton/neutron pairs in the arrangement of alpha particles in nuclei.

Conclusions

Although there is much left to be investigated the evidence so far indicates that

• All alpha particles which can be formed within nuclei are formed.
• There is some arrangement of those alpha particles into shells. The first shell involves two alpha particles. The second shell can accommodate twelve alpha particles. Third shell can accommodate at least eleven.
• Protons and neutrons are decidedly different in their effect on the binding energies of nuclides. Additional protons contribute very little to increased binding energy for nuclides unless they are combined with neutrons.

For a new theory in which the substructures are generally not alpha particles per se, but instead chains of alpha modules see Alpha Modules.