Waltzing with Magic Numbers

by Nicholas Mee on September 10, 2020

Marie Curie née Sklodowska

Marie Curie née Sklodowska became the first woman to win a Nobel Prize in Physics in 1903, sharing the prize with her husband Pierre Curie, and Henri Becquerel. She also became the first woman to win a Nobel Prize in Chemistry in 1911, and twenty-four years later her daughter Irène Joliot-Curie became the second woman to win the chemistry prize, shared with her husband Frédéric.

But who was the second woman to win a Nobel Prize in Physics?

Putting Your Ideas on the Table

Becquerel’s discovery of radioactivity and its further investigation by the Curies and others brought about a revolution in our understanding of matter. In 1911 Ernest Rutherford proposed that atoms consist of a tiny dense nucleus surrounded by a swarm of orbiting electrons. This helped clarify the meaning of the Periodic Table. Each entry corresponds to a different element and each element is composed of a different type of atom. The structure of the table reflects the chemical properties of these atoms and how they bond together to form molecules. And this behaviour is ultimately determined by how the orbiting electrons are organised in each atom.

The Periodic Table of the Elements

The Periodic Table encodes the arrangement of electrons in atoms. But what about the nucleus? By 1932 physicists knew that the atomic nucleus is composed of two types of particle—protons and neutrons. The chart below is known as a Segrè diagram after Italian physicist Emilio Segrè. Each square represents an atomic nucleus, where the number of neutrons (N) in the nucleus is plotted vertically and the number of protons (P) is plotted horizontally. 

The atom’s chemical identity is dictated by the number of protons in its nucleus. This equals the number electrons in a neutral atom, which determines the chemical properties of the atom. All the entries in a single column of a Segrè diagram are nuclei with the same number of protons, but differing numbers of neutrons. So they are the various isotopes of a single element. 

The Valley of Stability

The Segrè chart is colour coded to indicate a key feature of each nucleus, how long it hangs around before undergoing some sort of radioactive decay. The black squares show the stable nuclei and the deep red squares are long-lived, but unstable. Such nuclei are clustered into a narrow band known as the valley of stability. On either side of the valley are blue-coloured squares indicating highly unstable nuclei that decay very rapidly. The colour key for the half-lives of the nuclei is shown on the right of the chart.

The line labelled N=P indicates nuclei with equal numbers of neutrons and protons. The lightest stable nuclei lie on this line. These nuclei are some of the most important and abundant in the universe, including helium-4, carbon-12, nitrogen-14 and oxygen-16.

Heavier nuclei require a higher proportion of neutrons for stability, so going upwards the valley of stability veers to the left into the region where neutron number is greater than proton number. This is because protons are positively charged, so there is an electrostatic repulsion between them. In a sense, the glue from additional neutrons is required to compensate for the increased repulsion due to extra protons. Eventually, when the proton number passes eighty-two no quantity of neutrons is sufficient to maintain stability and the black squares peter out. There are, however, nuclei with as many as ninety-six protons that have lifetimes of several million years, producing a few more deep red squares in the upper reaches of the chart.

Magic Numbers

Some nuclei have more binding energy, and therefore greater stability, than expected from nuclear trends. This manifests itself in various ways and has numerous important physical consequences. Hungarian physicist Eugene Wigner noted that these nuclei contain particular magic numbers of protons, neutrons or both. The magic numbers are 2, 8, 20, 28, 50, 82, 126. 

The energy of the sun and stars is generated by the creation of the doubly-magic helium nucleus out of hydrogen nuclei.

I will quickly run through some of the consequences of the enhanced stability conferred by these nuclear magic numbers.

Magic nuclei are more abundant than other nuclei of comparable size. For instance, helium-4, which is doubly magic, as it is composed of two protons and two neutrons, is the most abundant compound nucleus in the universe.

Fusion of hydrogen into helium is what powers the sun and other stars for the bulk of their life span. Because helium is so tightly bound its formation releases far more energy than other fusion processes, and its stability inhibits the creation of heavier nuclei in stars. So heavier elements such as carbon and oxygen are only created in the cores of giant stars towards the end of their lives.

The oxygen-16 nucleus is also doubly-magic containing eight protons and eight neutrons and consequently oxygen is the third most abundant element in the universe after hydrogen and helium.

Nickel-56 is another doubly-magic nucleus composed of 28 protons and 28 neutrons. It lies very close to the absolute peak of nuclear stability, and is created in vast quantities in supernova explosions. Most of the light from a supernova is due to the radioactive decay of nickel-56 into cobalt-56, which subsequently decays to iron-56.

Radioactive nuclei decay step by step forming decay series that end when a stable nucleus is reached. All such decay series end with nuclei containing magic numbers of protons, neutrons or both. Element number 82 is lead, so its nuclei all contain a magic number of protons. Three decay series end with lead nuclei: lead-206, lead-207 and lead-208, the last of which is doubly-magic as it contains 126 neutrons. Until recently the endpoint of the other decay series was considered to be bismuth-209, containing a magic number of 126 neutrons. Strictly speaking this is no longer the endpoint as bismuth-209 is now known to decay, but very slowly. It is now regarded as quasi-stable, with a half-life vastly longer than the age of the universe, which makes the doubly-magic nucleus lead-208 the heaviest stable nucleus.

Finally, elements with magic numbers of protons, such as calcium, nickel and tin, with 20, 28 and 50 protons respectively, have an unusually large number of stable isotopes, far greater than their neighbours. For instance, element-50 tin has ten stable isotopes, more than any other element (the next most is seven), and far more than its neighbours. Element 49 indium has a single stable isotope indium-115 and element 51 antimony has two stable isotopes.

The atomic numbers of the noble gases are are 2, 10, 18, 36, 54, 86.

A Noble Analogy

So how do we account for the magic? There is a striking similarity between the extra nuclear stability of magic nuclei and the chemical stability of the noble gases—helium, neon, argon, krypton, xenon and radon. 

Electrons occupy the lowest energy states that are available. In an atom these states are the electron orbitals. They come in clusters with a similar energy known as shells, and between the shells there are energy gaps. Noble gas atoms have exactly the right number of electrons to completely fill several shells, and this accounts for their physical properties. Helium atoms contain two electrons, neon atoms contain ten electrons and so on. Such atoms are particularly stable and therefore unreactive, so they usually exist as individual atoms rather than bonded together into molecules. We can read the atomic numbers of the noble gases from the rightmost column of the Periodic Table. They are 2, 10, 18, 36, 54 and 86. We can think of these numbers as the atomic magic numbers. But clearly, they are not the same numbers as those found in nuclear physics.

Nuclear Shells

The forces operating within the nucleus are rather different to those that bind electrons in atoms, so we cannot expect the magic numbers to be exactly the same in the two cases. Nonetheless, nuclear magic numbers do represent the number of states available to nucleons in a complete set of nuclear shells. But it is one thing to argue by analogy, and another thing entirely to construct a physical theory that accounts for the nuclear magic numbers and makes further testable predictions.

Maria Goeppert-Mayer

Maria Goeppert-Mayer (1906-1972) was a German-born theorist who worked with Edward Teller on the American nuclear weapons programme at Los Alamos. From 1946 she worked at the University of Chicago and the nearby Argonne National Laboratory.

Goeppert-Mayer investigated the origins of the nuclear magic numbers. The shell model of atomic electrons offered her a clue about how to proceed, but understanding the nucleus is far more difficult than calculating electron energy levels in an atom. Goeppert-Mayer’s initial models matched the first few magic numbers, but failed for the higher magic numbers.

Enrico Fermi—one of the greatest physicists of the twentieth century—was based at the University of Chicago. Fermi had a remarkable insight into physics, and Goeppert-Mayer later recalled that it was a passing remark of Fermi’s that provided the hint she needed to solve the problem. Fermi asked whether there was any indication of spin-orbit coupling in the nucleus. 

Spin-orbit coupling. The energy level occupied by the nucleon on the left whose spin has the same orientation as the nucleus is lower than the energy level occupied by the nucleon on the right whose spin has the opposite orientation.

Fermi’s question might sound rather cryptic. He was asking whether there was a tendency for nucleons to align their spin with the overall spin of the nucleus. In other words, do nucleons that spin with the same orientation as the nucleus have lower energy than those that don’t? (See the image above.) It turns out they do and this was key to Goeppert-Mayer’s solution. The spin-orbit effect reorders some of the nuclear energy levels and this alters the number of nucleon states available in the higher energy shells. With her revised model Goeppert-Mayer could account for all Wigner’s magic numbers.

Goeppert-Mayer explained the spin-orbit effect quite poetically:

Think of a room full of waltzers. Suppose they go round the room in circles, each circle enclosed within another. Then imagine that in each circle, you can fit twice as many dancers by having one pair go clockwise and another pair go counter-clockwise. Then add one more variation; all the dancers are spinning, twirling round and round like tops as they circle the room, each pair both twirling and circling. But only some of those that go counter-clockwise are twirling counter-clockwise. The others are twirling clockwise while circling counter-clockwise. The same is true of those that are dancing around clockwise: some twirl clockwise, others twirl counter-clockwise.

Goeppert-Mayer published her explanation of nuclear magic numbers in 1949. Later that year an identical model was published by three German physicists Johannes Hans Jensen, Otto Haxel, and Hans Suess.

In 1963 Goeppert-Mayer became the second woman to win the Nobel Prize for Physics, sharing half the prize with Jensen,

for their discoveries concerning nuclear shell structure.

The other half of the award went to Eugene Wigner for his many contributions to atomic and nuclear physics.

Further Information

There is more about Emilio Segrè in a previous post: Scrap Metal from the Proton Merry-Go-Round

The discovery of the noble gases is discussed in this post: William Ramsay’s Noble Quest

There is more about radioactive decay series in this post: Frederick Soddy and the World Set Free

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