Atomic Nuclei

At the heart of every atom lives a nucleus. This nucleus, from what we understand, consists of protons and neutrons bound tightly in some unknown arrangement by powerful forces. Science books often draw an atom’s nucleus like this:

Looking like a cross between a raspberry and blackberry, this model shows protons (red) intermixed with neutrons (black) [1]. For light elements the number of protons and neutrons are in roughly equal proportions. As we move down the periodic table toward heavier elements we notice that neutrons steadily outnumber protons. The reason, it’s assumed, is that more neutrons are required to overcome the repulsive Coulomb force between larger numbers of protons.

Of course the above model is not intended to represent a true picture of a nucleus, although it seems that most texts infer a nucleus to be packed in this manner. Nonetheless I would suggest this ball-shape is unlikely. The problem is that putting protons into a ball will greatly enlarge the extent of their repulsive force. As can be clearly seen, a single neutron per proton will be insufficient to isolate the protons from each other.

Looking for clues

We know little about a neutron’s structure. It is impossible to see anything that tiny using present technology. Therefore we must look for clues that can be combined with our understanding of electrical forces in order to piece together a picture of the nucleus.

The best clues come from studying isotopes, i.e. nuclei with identical numbers of protons but varying numbers of neutrons. By looking at what isotopes are possible, their stability, half-life and decay products, we might learn how the protons and neutrons are arranged within.

Higher order nuclei

We’ll start by reviewing the models we have thus far for a proton, neutron and deuteron (a ‘heavy hydrogen’ nucleus: ²H).

These models appear stable and represent the simplest forms of the hydrogen nucleus and its components. How would we proceed in building the nucleus of higher elements, such as helium, lithium, etc., and various isotopes?

Notice the deuteron has a down-quark protruding at either end. A logical step would be to attach a proton to one of these. This would give the following:

What we now have is a Helium-3 nucleus, consisting of two protons and a neutron. Helium-3 is known to be a stable isotope of Helium, and from a visual perspective at least, it does appear stable. Just as with deuterium, this Helium-3 structure will need to rotate in order to hold the up-quarks at an altitude. We can also draw it in a flattened two-dimensional perspective:

Here the up-quarks are shown in the same plane, although they really alternate at right angles.[Note: particles are not drawn to scale in either 2D or 3D models.]

What’s next? There are still two down-quarks at either end. Could we attach a proton to one of them? If this could be done it would produce the isotope Lithium-4: 3 protons plus one neutron. Lithium-4 is known to exist but is very unstable. Attaching the proton to the right would yield:

There are a large number of up-quarks close together here, unlike the Helium-3 which spread them apart. Notice that when deuterium was converted to Helium-3, the up-quarks bent away from the centre, making it difficult to add further protons. Thus we would expect Lithium-4 to decay into Helium-3 by rejecting that end proton, which is what happens (proton emission).

[Note: for convenience I will be placing P and N above the added protons and neutrons respectively.]

Adding more protons to either end of this chain will produce higher order isotopes (Berillium-5, Boron-6, etc), but each of these would be (and are) even more unstable. So we’ll return to the Helium-3. Notice the proton on the right. It would seem that we could insert an electron within it. This would give:

This effectively converts the right-hand proton into a neutron, making the isotope Hydrogen-3. We know that Hydrogen-3 is largely stable with a half-life of 12 years and that it decays into Helium-3 by ejecting an electron (beta decay). According to the above diagram, all this does seem likely. The right-hand proton is wide enough to accommodate an electron. Let’s then try adding another neutron:

This will make Hydrogen-4. Unlike Hydrogen-3, Hydrogen-4 is very unstable. It has a half life of 10^-22 seconds and decays by emitting neutrons. In the above diagram we note that the neutron is being stretched horizontally due to an excess number of negatively charged particles (electrons and down-quarks) close together. Therefore it is quite likely that this additional neutron will be rejected as a whole. Adding additional neutrons will make Hydrogen-5,6 and 7, all of which are unstable and decay by ejecting those neutrons.

Going back to Hydrogen-3, a more suitable choice would be to add a proton:

This gives us Helium-4, which is of course very stable. The diagram appears to confirm this. The protons are spaced by a neutron and that partly unstable Hydrogen-3 electron is now better secured by the additional up-quarks.

Going forward, at this point we have a couple of choices: add another proton or another neutron. Adding a proton would yield Lithium-5; and a neutron, Helium-5:

Both Lithium-5 and Helium-5 are known to be unstable, and the diagram appears to confirm this. For Lithium-5 there are too many protons (or more specifically, up-quarks) close-together and too few negative charges to secure them; hence there would be much repulsion. As might be expected, Lithium-5 decays via proton emission into Helium-4.

Looking at the diagram of Helium-5, we might expect this to be stable; especially as Hydrogen-3 was fairly stable with a neutron attached to the end. But Helium-5 decays very quickly via neutron emission into Helium-4. Perhaps this is because trailing neutrons lean further to the right as the chain grows in length.

But if we add a proton to the end of Helium-5 we can un-distort and stabilise the neutron:

This gives us Lithium-6, which is stable.

Nuclear Magnetic Resonance

I could go on in this manner but you get the basic idea. By adding protons and neutrons to the end of the chain we can make different elements and isotopes. Certain isotopes will be stable and others not. Stability will depend on how close protons get to each other and how compressed the electrons and down-quarks are. For the unstable isotopes we can also predict what sort of decay mechanisms might occur.

It should be clear that this model is very different from the ball model of a nucleus and shows it growing into a chain of increasing length. Could a nucleus really be strung out like this? A big clue comes from nuclear magnetic resonance (NMR) which tells us that nuclei can be aligned with a magnetic field. A chain model seems highly appropriate for this. The orbiting up-quarks are all rotating in the same direction. Much like electrons circling in coils about an electromagnet, these up-quarks form a miniature solenoid.

It is easy to see how these mini ‘bar magnets’ can align themselves with an external magnetic field. Nucleons (protons and neutrons) arranged in a ball could also align themselves but in the process they would repel each other; similar to how it is difficult to align two nearby bar magnets in the same direction.

Large nuclei

For small elements the number of neutrons and protons are roughly equivalent but as the elements become heavier we notice a steady increase in the number of neutrons versus protons. The reason suggested for this is that extra neutrons help minimise repulsion between protons. If we look at the chain model, this is most likely true. But here the neutrons do more than act as ‘spacer washers’ between protons; they also provide electrons upon which the protons can secure themselves.

Another interesting fact is that, for larger elements, the ratio of neutrons to protons is usually around 1.5 but never over 2. This indicates that if their protons and neutrons were spread evenly over a chain, there would either be one or two neutrons between a pair of protons, but never three neutrons in a row.

Here’s how two versus three neutrons would look sandwiched between protons:

As can be seen, when there are two neutrons it appears the protons can attach to their outer negative charges (down-quarks and electrons) thus holding them stable. When there are three neutrons, the middle neutron is difficult to secure and there is also a large number of negative charges close together. Hence the three-neutron situation is unlikely to be stable.

Rules

Certain rules seem to govern the formation of stable isotopes.

1. A proton cannot be attached directly to another proton on the end of a chain.

2. A neutron cannot be on the end of a chain.

3. No more than two neutrons can lie in sequence.

4. For large elements, more neutrons are required, and subject to rule 3.

Helium-3 seems to violate rule 1. However its right-most added proton is not necessarily attaching to another proton because the two nucleons of deuterium are symmetrical and either could be considered a proton or neutron. Hydrogen-3 violates rule 2, as it is mostly stable. This stability is possible due to the Helium-3 proton widening itself sufficiently for the electron to be inserted.

Regarding rule 1, this only refers to protons at the end of a chain, not within a chain because a pair of protons can be secured by surrounding neutrons.

Adding to rule 3, these two neutrons in sequence should not be near the end of a chain because there are not enough surrounding protons to hold them in place. I.e. sequenced neutrons need to be embedded well within the chain. This would indicate why smaller elements are not allowed to have sequenced neutrons because their chain is not long enough.

Mysteries

The preceding observations don’t fully explain all isotopes. For example Helium-8 is more stable than Helium-7, and Lithium-11 more so than Lithium-10. In both cases the larger isotope has more neutrons and we would expect less stability. In the case of lithium, an interesting study [2] found the size of Lithium-11 and Lithium-12 nuclei was much greater than Lithium-10. It’s possible that additional neutrons have somehow separated from the rest of the chain and are floating at some distance. For example Lithium-11 may be made of a Lithium-9 and two neutrons held at some distance. How this could happen is unclear.




[1] Actually this red/black colouring is dated. Nowadays the trend is to use red/blue, with blue being neutrons. Red/white is also popular.
[2] http://admin.triumf.ca/docs/reports/annual_financial_admin2008/StarDust.pdf , page 2.