The Hoyle State

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

2, 8, 20, 28, 50, 82, 126, …

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

7.65 MeV – 7.27 MeV = 0.38 MeV

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

7456.89 MeV + 3727.38 MeV = 11184.27 MeV

This is more than the ground state energy of 12C, which—I said a while back—is 11,177.93 MeV. How much more?

11184.27 MeV – 11,177.93 MeV = 6.34 MeV

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.

9 Responses to The Hoyle State

  1. Harvey Brown says:

    Another twist to the carbon story. Until relatively recently, the half-life of carbon-14 was anomalous from a theoretical point of view. It is many orders of magnitude larger than both the half-lives of isotopes of other light elements undergoing the same decay process, and what standard calculations for nucleon-nucleon interactions in Gamow-Teller beta decay would indicate. In 2011, Maris et al. showed for the first time, with the aid of a supercomputer, that allowing for nucleon-nucleon-nucleon interactions in a dynamical no-core shell model of beta decay explains the long lifetime of carbon-14. Phys. Rev. Lett. 106, 202502 (2011).

    • John Baez says:

      Neat! That paper can be nabbed without a subscription from ResearchGate:

      • P. Maris, J.P. Vary, P. Navratil, W. E. Ormand, H. Nam and D. J. Dean, Origin of the anomalous long lifetime of 14C.

      Abstract. We report the microscopic origins of the anomalously suppressed beta decay of 14C to 14N using the ab initio no-core shell model (NCSM) with the Hamiltonian from chiral effective field theory (EFT) including three-nucleon force (3NF) terms. The 3NF induces unexpectedly large cancellations within the p-shell between contributions to beta decay, which reduce the traditionally large contributions from the NN interactions by an order of magnitude, leading to the long lifetime of 14C.

  2. Keith Harbaugh says:

    “I’ve had to watch seemingly intelligent scientists cling to these recent theories despite their lack of success.”

    Can’t resist providing a link to Sabine Hossenfelder’s video “Particle Physicists Continue Empty Promises”:

    She recalls some of the claims made for what various expensive projects would reveal about nature, and suggests advocates for these projects over-promised and under-delivered. Any thoughts on the issues she raises?

    In the same vein, I would ask just what we get for the $40G/year that goes to the NIH.

    As for me, I just wish the six parts of the wish list of the fusion researchers, helpfully listed here:
    https://www.sciencemag.org/news/2020/12/us-physicists-rally-around-ambitious-plan-build-fusion-power-plant
    could be fully funded. There are so many unknowns WRT making nuclear fusion power generation possible, it’s absolutely imperative we start reducing the unknowns. So many unknowns about what will actually happen in ITER when they finally turn it on. And that will take some serious govt funding.

    • John Baez says:

      Keith wrote:

      She recalls some of the claims made for what various expensive projects would reveal about nature, and suggests advocates for these projects over-promised and under-delivered. Any thoughts on the issues she raises?

      I have a lot of thoughts on this. To a zeroth approximation I agree with Sabine. After I quit work on quantum gravity I wrote this:

      What I’ve changed my mind about: should I be thinking about quantum gravity?, Edge, 2008.

      When I started trying to develop ‘green mathematics’ I wrote this, which is a kind of manifesto, urging mathematicians and physicists to focus on developing the ideas we need to adapt to life on a finite-sized planet:

      Network theory (part 1), Azimuth, 4 March 2011.

      On Monday March 8th I’ll give a talk called Theoretical physics in the 21st century, in which I’ll try to say more!

    • Any thoughts? She has some good points, but throws the baby out with the bathwater. Also, sometimes I get the impression that she thinks that since she has written a paper on something, or even just mentioned it in a blog, then the case must be closed.

      I reviewed her Lost in Math book as a guest post on her blog, which she was kind enough to allow even though the review was not wholly positive. There, my criticism is that she misunderstands much of the discussion about fine-tuning in physics. My next paper, almost ready to go, will deal with that. (First things first; I’ve had 5 single-author papers accepted in the last 13 months, even though until the end of December 2020 I had been working full-time outside of academia.)

    • Keith Harbaugh says:

      Sabine Hossenfelder’s video “Particle Physicists Continue Empty Promises” linked to above appeared on 2020-10-22.
      It appeared in a certain context, explained in a 2020-07-02 article
      “Europeans Decide on Particle Strategy”
      https://physics.aps.org/articles/v13/105

      From the physics.org article:

      “CERN needs to have a project for after the LHC,” says Halina Abramowicz, chair of the European Strategy Group, from Tel Aviv University in Israel.

      The main objective of any post-LHC endeavor will be to look for new particles or phenomena that go beyond the standard model of particle physics.
      Physicists are still in the dark as to what this “new physics” will be, so the best way forward is to study the Higgs boson with greater precision, Abramowicz says.
      The Higgs is unique in that it should interact with all particles, even ones that physicists haven’t detected yet.
      “The Higgs does not differentiate: if there is something out there, it will couple to it,” Abramowicz explains.

      • While one can discuss how to best spend a finite amount of money, I’m reminded of something which Robert Pirsig wrote: The television scientist who says „Our experiment is a failure; we didn‘t find what we were expecting“ is suffering mainly from a bad scriptwriter. If you know what you will find, there is little need to do the experiment. General-purpose machines are very important. Many interesting things have been found by such things which were not even though of when they were built, much less used to justify the funding.

  3. John Baez says:

    Some of my remarks about the Hoyle state were wrong, and I fixed them. Here’s the fixed version, in case you already read this article:

    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

    7.65 MeV – 7.27 MeV = 0.38 MeV

    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 2 alpha particles. 12C, as we’ve seen, can be made from 3. 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

    7456.89 MeV + 3727.38 MeV = 11184.27 MeV

    This is more than the ground state energy of 12C, which—I said a while back—is 11,177.93 MeV. How much more?

    11184.27 MeV – 11,177.93 MeV = 6.34 MeV

    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.

  4. domenico says:

    I am thinking that in the nucleus the u-d quark bond could exist, like in the Kekulé structure with a linear combination of nuclear wavefuction.
    For example there are excited state for the neutron, and proton, where the quarks are aligned in delta barions

    https://www.researchgate.net/figure/Qualitative-picture-of-the-neutron-left-and-0-right-charge-distributions-in-the_fig2_2173550

    so that a quark in the neutron could exchange a pion to change d ->u in a proton and it seem work.
    The deuterium could be formed by a neutron n(d-u-d) and a delta(u-d-u) baryon with three interaction in the linear configuration of quark: the energy of the neutron and delta is 939.6+1232=2171.6 Mev, the energy of the neutron and proton is 939.6+9383=1877.9 Mev, so that the three bonds have an energy of 97,9 Mev.
    This could be true for the tritium, using three linear quark configuration 2n(d-u-d) delta(u-d-u) with six possible bonds. The energy of the 2n+delta is 2939.6+1232=3111,2Mev, the tritium energy is 3,0160492931,494102=2809,5Mev so that the total energy of the bonds is 301.7Mev, the energy of a single bond is 301.7/3=100,6Mev (the pion interaction change the quark type, so that there can be only one interaction between nucleons at a time).
    The alpha particle is 2n(d-u-d) e 2p(u-d-u), if there is an interaction between linear conformation of quarks then 2n(d-u-d) interact with 2delta(u-d-u), the alpha energy is 3727.4 Mev, the energy of the nucleons and barions is 2’939,6+21232=4343.2Mev, so that the difference in energy is 615,8 Mev and the energy of a single bond is 615.8/6 = 104.8 Mev.
    Now I am evaluating the energy of the bounds of nucleon in the triangular conformation (delta barions with triangular configuration of quarks and protons) to obtain a magical number using an octahedron with two quark in each vertex; if everything work the magical number could be a geometric result.

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