What are the 'magic numbers' in nuclear physics, and why are they so powerful?

Some atoms are stable, while others seem to fall apart. Lead-208 will probably last forever, while the synthetic isotope technetium-99 exists for just hours. The difference lies in the structure of the atom's nucleus, with certain "magic numbers" of nuclear particles making some isotopes especially resistant to radioactive decay.

So what are these magic numbers, and why are they so special?

The stability of atomic nuclei varies wildly with the number of nuclear particles they contain. Some, like lead-208 and calcium-40, have been around since Earth first formed. Known as primordial isotopes, they will likely survive until the end of time. Others, like oganesson-294 and tennessine-294, are lost to radioactive decay in an instant, with fleeting half-lives of just 0.89 and 0.80 milliseconds, respectively.

This stability seems partly connected to the mass of the atom, with heavier elements proving less stable. But in the 1940s and '50s, scientists observed that many of the lighter elements also had radioactive isotopes; both carbon-14 and potassium-40 undergo radioactive decay slowly and are responsible for much of the planet's background radiation.

Intriguingly, these scientists noticed that very particular numbers of protons and neutrons appeared to result in unusually stable nuclei, and these values became known as magic numbers.

"The magic numbers are 2, 8, 20, 28, 50, 82 and 126," said David Jenkins, a nuclear physicist at the University of York in the U.K. "If you take the lightest one — two protons and two neutrons — that's the nucleus of the helium atom, and we know that's a very stable combination of protons and neutrons."

Related: Why isn't an atom's nucleus round?

Shell game

Helium nuclei, also known as alpha particles, are spontaneously emitted from heavier, unstable atoms as they undergo nuclear decay.

"If you think about it, that's very weird," Jenkins said. "If an atom is going to decay, why doesn't it lose protons or neutrons one at a time? The reason is that the alpha particle is very very stable, and that's related to this idea of magic numbers."

Other magic nuclei include oxygen-16 (eight protons and eight neutrons), calcium-40 (20 protons and 20 neutrons) and lead-208 (82 protons and 126 neutrons), the heaviest stable element known.

To understand these bizarre observations, physicists proposed the "nuclear shell model," which draws parallels with the electronic shells used to explain the chemical behavior of atoms.

"The idea was that protons and neutrons sit in shells, a bit like the electrons in an atom, and nuclear excitations would involve protons and neutrons jumping up and down between those shells," Jenkins explained.

Like their electron analogues, these nuclear shells have fixed energy values known as quantized states, and the system is most stable when these shells are completely filled. The exact reasoning behind this is a complex combination of quantum mechanical factors, but it's thought that the strong force — the fundamental interaction that holds the protons and neutrons together in the nucleus — is higher than expected per particle in completed shells.

Magic numbers are therefore simply the numbers of particles required to fill each of these nuclear shells, with separate levels for protons and neutrons. Individual isotopes can correspondingly be singly magic, with a magic number of either protons or neutrons (for example, the primordial isotope iron-56), or doubly magic, with magic numbers of both protons and neutrons (like oxygen-16 and lead-208).

These doubly magic systems are few and far between, but they possess some intriguing quantum properties, Jenkins said.

"The doubly magic systems have a spherical distribution of matter and charge" — a completely round nucleus, he said. "Most nuclei are deformed and rotate. They have a very different structure."

No one knows how far this model will stretch. Tin-100 — the heaviest doubly magic nucleus, with 50 protons and 50 neutrons — has a half-life of just 1.2 seconds, while unbihexium, the next magic element after lead, has never been synthesized. Therefore, whether this magic stability boost will be enough to allow scientists to add an eighth row to the periodic table remains an open question.

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