We have been enthralled by a variety of magic shows in our childhood. However, this article isn’t going to discuss any of those magic tricks. Rather, it will shed some light on the reason behind the magic of nuclear stability: The magic numbers!

Magic numbers for atoms and nuclei

In our junior chemistry classes, we have learned that those atoms with filled or half-filled atomic orbitals are the most stable ones. For instance, noble gases having filled atomic orbitals are the most stable of all the elements. So in the case of atoms, the number of electrons in atomic orbitals defines the stability. Therefore, the magic numbers for atoms are 2, 10, 18, 36, 54, and 86, corresponding to the total number of electrons in filled electron shells. A stable element requires a great amount of energy to react with other elements, and that is why noble gases are the least reactive.

Now coming to nuclei, it has been observed that the nuclei of some atoms having a specific number of nucleons (protons or neutrons) are much more stable than other nuclei. Such atomic nuclei have a comparatively higher average binding energy per nucleon and are thus more stable against nuclear decay. So in nuclear physics, a magic number refers to the number of nucleons (either protons or neutrons) such that they are arranged into complete shells within the atomic nucleus.

Magic Numbers in Nuclear Physics

The seven most widely recognized magic numbers are 2 (helium), 8 (oxygen), 20 (calcium), 28 (nickel), 50 (tin), 82 (lead), and 126 (hypothetical unbihexium). Amongst these, 2, 8, 20, 28, 50, and 82 corresponds to the number of protons,  while 126 is the only known magic number associated with neutrons.

Explanation of nuclear magic numbers: The Shell Model

After World War II, the study of nuclear structure and magic numbers became a trending research topic. In 1937, Neils Bohr and F. Kalcar proposed the liquid drop model of the nucleus, where the atomic nucleus was compared to a liquid drop. Although this model was of utmost importance to understand some of the basics of binding energies, it could not explain why only some specific nuclei having protons or neutrons or both as 2, 8, 20, 28, 50, 82, 126 (magic numbers) have higher binding energies, making them more stable than others. This led scientists to find a better model for enhanced explanations. This is where Maria Mayer’s shell model came into the picture.

The 7 Magic Numbers And Their Importance In Nuclear Physics. 1

Maria Goeppert Mayer Courtesy of the Sarah Lawrence College Archives

Dmitry Ivanenko first proposed the shell model in 1932, and later Maria Mayer and some other physicists developed it to understand the ambiguity about magic numbers in 1949. The nuclear shell model is partly analogous to the atomic shell model, which describes the arrangement of electrons in an atom so that a filled shell results in greater stability.

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According to the shell model, the motion of each nucleon inside the nucleus is governed by the average attractive force of all other nucleons. As a result, the orbit of motion is determined by a potential energy function V(r), representing the average of all interactions with other nucleons and is the same for each particle. The resulting orbits of motion form “shells,” just like the orbits of electrons in atoms. Moreover, the potential term V(r) is analogous to the Coulomb potential term in the atomic shell model.

As the nucleons are arranged in shells inside the nucleus, each shell can allow only a limited number of nucleons as permitted by the Pauli exclusion principle. However, when all of the possible sets of quantum numbers available are occupied and the shell is completely filled, it results in a special configuration that is particularly more stable and usually has low energy.

Nuclear shell model - magic numbers
Magic numbers as explained by the shell model

The shell model proved to be instrumental in explaining the existence of magic numbers and the stability and high binding energy based on closed shells. Moreover, it also helped Maria Mayer to step up to the podium in Stockholm, Sweden, to accept the Nobel Prize in Physics in 1963.

Doubly magic nuclei

Some nuclei exist in which both the neutron number and the proton (atomic) number are equal to one of the magic numbers. These kinds of nuclei are termed doubly magic. Some known doubly magic isotopes include helium-4, helium-10,  oxygen-16,  calcium-40, calcium-48, nickel-48, nickel-56, nickel-78, tin-100, tin-132, and lead -208.  The doubly-magic effects allow the existence of those stable isotopes which otherwise would not have been expected.

Moreover, magic effects prevent unstable nuclides from decaying as rapidly as expected otherwise. It has been found that nuclei with even numbers of protons and neutrons are more stable than those with odd numbers. In our universe, magic number shell effects are seen in ordinary abundances of elements. For instance: helium-4 is among the most abundant (and stable) nuclei in the universe, and lead-208 is the heaviest stable nuclide.

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