Nuclear physics is complicated compared to atomic physics, because the strong force is complicated compared to the electromagnetic force, and nucleons—protons and neutrons—are bag-like groupings of quarks and gluons held together by the strong force. They resemble elastic bags that attract each other. They jostle each other in the nucleus… governed by the rules of quantum mechanics.
To begin to understand such a complex thing as a nucleus, people started with approximate models. In 1930 George Gamow introduced the ‘liquid drop model’, which was further developed by Niels Bohr, John Archibald Wheeler and Carl F. von Weizsäcker. The idea is to treat the nucleus as a droplet of an incompressible fluid with some surface tension—but again, quantum-mechanically.
Another model, more reminscent of atomic physics, is the shell model. Here neutrons are protons are treated as moving in a potential well (which is actually created by their interaction with each other). Since protons and neutrons each separately obey the Pauli exclusion principle, there are—approximately—separate shells for each kind of particle, which fill up when their reaches a so-called magic number
Thus, nuclei with a magic number of either protons or neutrons are especially stable, and ‘doubly magic’ nuclei, with a magic number of protons and a a magic number of neutrons, are even more stable: think of helium-4, oxygen-16, calcium-40, nickel-56, or lead-208 (with 82 protons and 126 neutrons).
Yet another interesting approximation is to think of a nucleus as made of smaller nuclei, especially these four:
• the deuteron: a deuteron, the nucleus of deuterium or 2H, is a proton and neutron stuck together.
• the triton: a triton, the nucleus of tritium or 3H, is a proton and two neutrons stuck together.
• the helion: a helion, the nucleus of 3He, is two protons and a neutron stuck together.
• the alpha particle: an alpha particle, the nucleus of 4He, is two protons and two neutrons stuck together.
Of these, all but the triton is stable on its own, and the triton has a half-life of 4500 days, which is essentially forever on the timescale at which particles in the nucleus do things. Compare a truly unstable nucleus like 4H, consisting of one proton and three neutrons: this has a half-life of 1.39 × 10-22 seconds.
Here’s a great example of how it can sometimes be useful to think of atomic nuclei as assemblages of deuterons, tritons, helions and alpha particles. The nucleus of ordinary carbon, 12C, consists of 6 protons and 6 neutrons. But it has an excited state—a state of higher energy—in which it acts like 3 alpha particles orbiting each other! This is called the Hoyle state.
The Hoyle state has energy 7.65 MeV more than the lowest-energy state of 12C. To get a feeling for how much that is, it helps to know that when a 12C is in its lowest-energy state—called its ground state—its energy is 11,177.93 MeV. So, the Hoyle state has just a tiny bit more energy than the ground state.
But here’s a better way to think about it. The ground state energy of 12C is 7.27 MeV less than that of 3 alpha particles. So, the Hoyle state of 12C has
more energy than 3 alpha particles. This means that carbon in its Hoyle state can break apart into 3 alpha particles! It can also decay back to the ground state of 12C. But it’s not a bound state: it’s not held together, it can fall apart into alpha particles.
But here’s the really interesting thing about the energy of the Hoyle state: it’s almost the same as the energy of a beryllium nucleus plus an alpha particle! An ordinary beryllium nucleus, 8Be, is made of 4 protons and 4 neutrons. Thus, it can be made from two alpha particles. 12C, as we’ve seen, can be made from three. But the rest energy of 8Be plus that of an alpha particle exceeds that of 12C, thanks to binding energy. In fact, that sum is closer to the energy of the Hoyle state!
Let’s see how it works. The ground state energy of 8Be is 7456.89 MeV. The ground state energy of an alpha particle is 3727.38 MeV. Summing them up, we get
This is more than the ground state energy of 12C, which—I said a while back—is 11,177.93 MeV. How much more?
On the other hand, we’ve seen the energy of the Hoyle state is 7.64 MeV more than that of 12C. These numbers are pretty close.
This coincidence is important, and it has a romantic history. The astrophysicist Fred Hoyle predicted its existence based on stellar evolution. Without a state of this sort, it’s unlikely that carbon would be formed when alpha particles smack into beryllium nuclei in a star! And that would be a serious roadblock to the formation of carbon.
This is sometimes counted a success of the anthropic principle, since without carbon there would be no life…
…. well, no life containing carbon anyway.
As far as I’m concerned, the anthropic twist is wholly unnecessary and distracting. You see carbon in stars, you know it must have gotten there somehow, and you guess there must be an excited state of carbon to explain this. It’s a perfectly fine piece of detective work. Why spoil it by tacking on the observation “and if there were no carbon, there would be no intelligent life!”
In fact Hoyle didn’t mention the anthropic business in his original argument in 1953: he was focused on the observed appearance of elements in stars. Only in 1965 did he add a remark that had the energy levels been different, “it is likely that living creatures would never have developed”. And only later, in 1979, did Carr and Rees claim that the prediction of the Hoyle state was, or could have been, a triumph of the anthropic principle.
For a careful dissection of how the mythology surrounding Hoyle’s prediction grew and grew over time, read this:
• Helge Kragh, Higher Speculations: Grand Theories and Failed Revolutions in Physics and Cosmology, Section 9.2: Anthropic Reasonings, Oxford U. Press, Oxford, 2011.
This is a truly wonderful book, though I find the failed theories of the 1800s inspiring, and the recent ones merely depressing, since for decades I’ve had to watch seemingly intelligent scientists cling to these recent theories despite their lack of success.
Anyway, what fascinates me about the Hoyle state is not the anthropic baloney, but the idea of a carbon nucleus as three alpha particles engaged in a complicated quantum dance!
Of course this is just a simplified picture, an approximation. For more, see these:
• Natalie Wolchover, A primordial nucleus behind the elements of life, Quanta, 4 December 2012.
• David Jenkins and Oliver Kirsebom, The secret of life, Physics World, 7 February 2013.
While these authors are unable to resist retelling the anthropic just-so story, their articles contain interesting details, simply explained, of how physicists are trying to get a better understanding of the Hoyle state.
Recent supercomputer calculations show that even carbon in its ground state is approximately described by three alpha particles, orbiting each other in a compact triangle! Carbon in its Hoyle state, on the other hand, is approximately a ‘bent arm’ configuration of three alpha particles:
This image is a modified version of one in the Physics World article by Jenkins and Kirsebom.
For something about experiments rather than computations, try this:
• Hans O. U. Fynbo and Martin Freer, Rotations of the Hoyle state in carbon-12, Physics, 14 November 2011.