Maria Goeppert Mayer: Nuclear Physics, Magic Numbers and the Onion Madonna

Maria Goeppert Mayer, linocut, 11" x 14", by Ele Willoughby, 2025

The first thing that came to mind when I thought about lead for the next #PrinterSolstice2425 prompt is how it is the end of many chains of nuclear transmutations. Lead is particularly stable. The woman who figured out why was the second woman to win the Nobel Prize for physics: German-American theoretical physicist Maria Goeppert Mayer (1906-1972, née Göppert). Another woman would not win the physics prize until Canadian Donna Strickland in 2018.

Maria Göppert was born in Kattowicz, a Silesian city then part of Prussia, now part of Poland, the only child of father, pediatrician and sixth generation professor Frederich Göppert and mother Maria (née Wolfe). Maria would grow up, proud to be a seventh generation academic. The family moved to Göttingen when she was four when her father got a position at the university. She was closer to her father, reasoning, he was more interesting, "He was after all a scientist." He encouraged her to aim for more than a life as a housewife. She went to schools for girls intending to pursue higher eduction. When her suffragette-run private prep school closed before she completed the three year program, she took and passed the university entrance examination at 17, a year early. She entered the University of Göttingen, where Emmy Noether was a professor, to study mathematics, in 1924, spending a term in Cambridge before completing her degree. The prestigious university was known for its world class math and physics departments. Renown mathematician David Hilbert was a neighbour and like famed physicists Max Born and James Franck, a family friend of the Göpperts. There were other female students but the others were studying to become mathematics teachers for girls. Göppert on the other hand became interested in physics and decided to pursue her doctorate with Max Born, himself later a Nobel laureate. In her 1930 thesis, she studied two-photon absorption by atoms - something which was virtually impossible to verify until 1961 with the invention of the laser. Year later, her fellow Nobel laureate Eugene Wigner called her thesis, "a masterpiece of clarity and concreteness" and today the two-photon absorption cross-section is named the Goeppert-Mayer (GM) unit.

She met American physical chemist Joseph Edward Mayer, a Rockefeller Fellow who was working for Göttingen physicist (and later Nobel laureate) James Franck, who boarded with the Göppert family. The two fell in love and married in 1930 after she completed her PhD. They moved to the United States, where he got a faculty job at John Hopkins, at the height of the Depression. They had two children: Maria Ann and Peter Conrad. Like many women scientists married to scientists, she found anti-nepotism rules prevented her from getting a professorship at John Hopkins, but she was hired as an assistant dealing with German correspondence. The job had a small salary, but allowed her access to the facilities and she was able to teach courses. She collaborated with her husband and with Karl Herzfeld, a fellow German theoretical physicist at John Hopkins, on applying quantum mechanics to the chemistry of organic molecules. She also made several visits back to Göttingen to collaborate with Max Born in 1931, 1932 and 1933, before the nazis came to power and Born and Franck lost their jobs. She and Herzfeld became involved with refugee efforts, horrified to see academics of Jewish descent driven out of university jobs. She became an American citizen in 1933. She published a landmark paper on double beta decay in 1935. In 1935 Edward Teller got a position at the nearby George Washington University and they would discuss developing theoretical physics. In 1937, her husband was fired; he believed this was because of Maria's presence in the lab and the dean of science's hatred of women. Herzfeld agreed, but also noted the anti-German sentiment that greeted him, Maria and Franck who were all now at John Hopkins. Mayer got a job at Columbia n 1937, where Maria took an unpaid position. Harold Urey and Enrico Fermi arrived at Columbia in 1939 and the three became good friends. In 1940, Joe and Maria published their textbook Statistical Mechanics. Fermi asked her to work on the valence-shell of transuranic elements, which she correctly predicted would form a new series similar to rare earth elements. In 1941 she was elected a fellow of the American Physical Society; the letter from APS was address, "Dear Sir," as if they had not considered an other possible greeting might be appropriate for fellows. She still had no salary, until later that year when she was hired by Sarah Lawrence College, initially part-time, to teach mathematics, physics, physical chemistry and general science courses. 

In 1942, she was recruited to work for the Manhattan Project. First she worked part-time for Urey at Columbia's Substitute Alloy Materials Laboratories trying to separate the fissile uranium-235 which could be used for weapons from natural uranium. She looked at uranium hexafluoride and investigated whether photochemical reactions could be used; while unfeasible in the 1940s, now lasers can be used to separate isotopes. Joe was working on conventional weapons at the Aberdeen Proving Grounds in Maryland five days a week. It was a challenging time for the family. The children had a nanny while their parents were occupied with war work. She was vehemently anti-nazi but found feared the consequences of producing a weapon which would harm friends and family in German if it were deployed there. She had taught her students at Sarah Lawrence that, "Man's scientific discoveries and inventions might very likely destroy him." She was stretched pretty thin and suffered several illnesses and on top of her full time teaching duties and childcare so negotiated reduced hours working for Urey. Teller recruited her to work on  the Opacity Project on the properties of matter and radiation at extremely high temperature; Teller was working on his "Super" bomb, which would become the basis of the H-bomb. Her husband Joe was sent to fight in the Pacific; Maria decided to leave the children in New York and join Teller's team at Los Alamos. When Joe came back early they returned together to New York in 1945. 

After the war her husband got a job at the University of Chicago; as did she, but it was as a voluntary professor of physics. When Teller joined the university she continued working on the Opacity Project and on the origin of elements. She was offered a part-time senior theoretical physicist job at the nearby Argonne National Laboratory. She responded that she didn't "know anything about nuclear physics" when offered the job. It was a shift in focus which laid the ground for her most impactful discoveries. She programmed the Aberdeen Proving Grounds ENIAC early computer to solve criticality problems of their liquid metal cooled reactor using Monte Carlo methods (that is, with statistical models you might use if trying to calculate gambling odds at the casinos in Monte Carlo). With Jacob Bigeleisen she derived the Bigeleisen-Mayer equation, also known as the Urey model (who independently arrived at the idea), of approximate isotope fractionation in isotope exchange reactions used in quantum chemistry and geochemistry. 

She began working on her mathematical model of the nucleus to explain why nuclei with certain numbers of nucleons (protons and neutrons) are more stable. As you grow nuclei from hydrogen (one proton) to the hundreds in transuranic elements by adding nucleons there are points where the binding energy of the next nucleon is a lot lower than the last and the nuclei are more stable and common in nature. The numbers of nucleons producing very stable nuclei: 2, 8, 20, 28, 50, 82 and 126 were dubbed "magic numbers" by Wigner. For protons these nuclei are helium, oxygen, nickel, tin, lead and the theoretical unbihexium. From her work on the origin of elements she recognized that these were all more common than their periodic table neighbours. You can also get nuclei with magic numbers of neutrons or "doubly magic" nuclei with magic numbers of protons and of neutrons. In 1932 Ivanenko and Gapon were the first to propose that perhaps nucleons were distributed in shells. But in 1937, Niels Bohr and  Kalcar proposed the useful "liquid drop" model of the nucleus. The liquid drop model helped physicist comprehend binding energies, and was for instance, how Lise Meitner and her nephew Otto Frisch explained nuclear fission. But the model did not explain the relative stability of different nuclei or the nature of magic numbers. Because Goeppert Mayer had not come from a nuclear physics background she was less biased in favour of the liquid drop model.

In 1948 she published her first paper summarizing the support for a shell model, but she did not yet have an explanation for the distribution of magic numbers. Fermi suggested she look at spin coupling, and she had a flash of insight. She realized that magic numbers could be explained by a nested sequence of closed nuclear shells where pairs of protons and neutrons would couple together.  In 1949, Maria Goeppert Mayer build her nuclear shell model (green diagram) where the magic of these numbers is explained by a nested sequence energy levels (determined by spin and angular momentum & like with electrons beyond the nuclei, the Pauli exclusion principe) of filled shells. Pauli himself called her “The Onion Madonna” for her model’s onion-like layers. 

In quantum mechanics nucleons have two possible spins: up or down. If you combine spin with their orbital motion you get total angular momentum. She’d found that if orbital and spin motions align to produce a maximum total angular momentum, nucleon energy level shifts down but when they go opposite directions nucleon energy level shifts up. The largest gaps between these shifted energy levels explain the “magic numbers” and these points mark the shells boundaries. Further, her model explained the ground state spins and magnetic moment of nuclei, for which there had been no previous explanation.

Very shortly thereafter, other physicists (Haxel,  Jensen & Suess) independently developed the same idea and & she started collaborating with them & co-authored a book: Elementary Theory of Nuclear Shell Structure with Jensen in ‘50.  In ’63, she & Jensen shared half the Nobel "for their discoveries concerning nuclear shell structure" with Wigner awarded other half.  In a letter, she addressed Jensen as "My Nobel Shell Brother."

She was finally appointed a full professor of physics at a major university in 1960, at UCSD, where her husband was also offered a position. She suffered a stroke but continued teaching and working despite health problems. In 1971 she suffered a heart attack that left her comatose and she died in February 20, 1972. 

The American Physical Society now awards the Maria Goeppert Mayer award to women in physics at the beginning of their careers. In 2018 they named Argonne National labs an APS Historic Site in recognition of her work. A crater on Venus has been named in her honour. In 1996 she was inducted in the National Women's Hall of Fame. The UCSD physics department is named Mayer Hall after her and Joe.

References

August, 1948: Maria Goeppert Mayer and the Nuclear Shell Model, APS News, August 1, 2008

Buntar, Simran, Maria Mayer - The First Woman to Win the Nobel Prize for Nuclear Physics, Secrets of the Universe blog,  accessed January, 2025

Maria Goeppert  Mayer, Wikipedia, accessed January, 2025

Maria Goeppert Mayer: Revisiting Science at Sarah Lawrence College, Sarah Lawrence College Archives Exhibit, accessed January, 2025

Landau, Elizabeth. The Last Woman to Win a Physics Nobel, Scientific American, September 26, 2017.

Magic number (physics), Wikipedia, accessed January, 2025

Nuclear shell model, Wikipedia, accessed January, 2025

Sachs, Robert G. Maria Goeppert Mayer, 1906-1972, National Academy of Sciences, 1979.

Harriet Brooks, Nuclear Physicist

Harriet Brooks, linocut 9.25" x 12.5" by Ele Willoughby, 2018

When physicists first stumbled upon the phenomenon of radioactivity and shockingly found themselves becoming modern-day alchemists, at the turn of the last century, a promising young Canadian woman made fundamental contributions to our understanding of the nascent field of nuclear physics. Ernest Rutherford (later Lord Rutherford the Nobel laureate) recruited Harriet Brooks (1876 - 1933) while he was working at McGill University in Montreal. Rutherford, now considered one of the giants of early 20th century physics, was a frank New Zealander, known for his dry humour who famously described all of science as either physics or "stamp collecting". I do not imagine he was generous with his compliments, but he stated that Brooks was second only to Marie Curie in her capacity for and understanding of radioactivity. An excellent research physicist, Rutherford was clearly a dedicated and thoughtful supervisor who knew talent when her saw it.

Born July 2, 1876, in Exeter, Ontario, Harriet was one of nine children. Her family was not wealthy and after her father's flour mill burned down and was not covered by insurance, occasionally food was a bit scarce. Only she and her sister Elizabeth, who both had an aptitude for mathematics, attended university. Harriet entered McGill in 1894, only 6 years after the university had admitted its first female student. She graduated in 1898 with a B.A. in mathematics and natural science just as Rutherford arrived from England. He took her on as his first graduate student. He was careful not to assign her a problem in the new and technically difficult field of radioactivity right away, because he felt it was important that young researchers have some appreciated and acknowledged success at the beginning of their carreers, or they might become mistakenly disappointed in their abilities. So after publishing her results in 1899 in the Transactions of the Canadian section of the Royal Society, Harriet completed her master's degree in 1901 on the "Damping of Electrical Oscillations," before embarking on radioactivity research. As a promising researcher, she was appointed a non-resident mathematics tutor for the new women's college Royal Victorian College at McGill in 1899.

Apparatus and schematics of their thorium emanations research, Cabinet B of the Rutherford Museum at McGill

Assigned to on the puzzle of the thorium "emanation" she discovered some of the earliest evidence of transmutation! At that time, physicists knew that some elements gave off alpha particles (which we now know are helium nuclei, two protons and two neutrons bound together), beta particles (which we now know are electrons) and gamma rays, but these "emanations" were different. They knew it might be a radioactive gas, vapour or even a fine powder, but she concluded that it must be a gas of lower atomic weight than thorium - something chemists of the day believed impossible. She co-authored the paper with Rutherford, "The New Gas from Radium" in 1901. This new gas was Radon. She went on to work on measuring the atomic mass of this new gas. Assigning the discovering of a new element to a given individual, especially at this heady time, is tricky, though history is often told much more simply. Some argue that Friedrich Ernst Dorn discovered the element, as he published that Radium compounds emanate a radioactive gas he named Radium Emanation (Ra Em) in 1900. This however, is merely an observation of the existence of a specific isotope of Radon. Rutherford and Brooks themselves credited the Curies for first observing the "emanation".  Today, many credit Rutherford with the "true" discovery of Radon as an element (i.e. a "New Gas"), though he himself was always careful to reference Brooks' work. Between his sole-authored Nature paper of 1901, and his towering reputation, Brooks' role in this discovery was been largely forgotten for decades and only recently, has her role been rediscovered.* While the complete story is messy, a strong case can be made that Harriet Brooks discovered Radon, a new element with a lower atomic mass. It is incontestable that she was amongst the first in the world to observe any form of the element and to attempt to measure its atomic mass.

This 2015 issue of the Canadian Journal of Physics had the cover caption:  Harriet Brooks, pioneering Canadian nuclear physicist, discovered radon in 1901. Her Ph.D. supervisor, Ernest Rutherford, compared her to Marie Curie. On the faculty of Barnard College, she was forced to resign when she became engaged, as marriage was forbidden for female faculty, a great loss to physics (photo courtesy of Wm. Notman & Son; II-123880, McCord Museum).

Brooks proceeded to pursue a doctorate at the famous woman's college Bryn Mawr. She won the President's European Fellowship to go spend 1902-1903 in Cambridge working with (future Nobel laureate) J.J. Thomson. Thomson lacked Rutherford's dedication to guiding young researchers and largely left her on her own. She, like many, especially women, before or since, suffered from impostor syndrome and described herself as "a terrible bungler in research work" in a letter to Rutherford. She wildly underestimated her own skills and accomplishments and was hindered by Thomson's erroneous belief at the the time that radioactivity was a chemical process.** Nontheless, Brooks she made the first measurement of the half-life of Radon (her value: 1 minute, versus the modern value of 55 seconds) while working in Thomson's lab.

Instead of returning to Bryn Mawr to complete her doctorate, she went back to McGill to work with Rutherford for a year. Her biographers Marelene Rayner-Canham and Geoff Rayner-Canham suggest this may have due to her loss of confidence. While back at McGill, she noticed what she called the 'volatility' of radioactive substances, and how a non-radioactive plate placed in a radioactive container would become radioactive. She was seeing the first evidence of her discovery of atomic recoil. When the radioactive element emitted an alpha particle in one direction, the daughter nuclei would be propelled in the opposite direction - sometimes with such force that they became embedded in the plate (which hence becomes radioative). This method was later used by Otto Hahn and Lise Meitner to separate daugther nuclei and identify new elements. Hahn (who alone was granted the Nobel for the discover of nuclear fission he and Meitner made)*** claimed to have discovered atomic recoil, but Rutherford wrote to him to point out it was in fact Brooks. During this time she also charted the decays of Thorium, Radium, and Actinium and discovered not only did radioactive elements transmutate into new elements, but that these products in turn decayed, laying the groundwork for the discovery of nuclear decay series. She published her results in 1904.

One of the important examples of a nuclear decay series showing how various Uranium isotopes can transmutate in a series of reaction (i.e. to Thorium to Radium to Radon...)

In 1904, she became a physics tutor at Columbia's women's college Barnard College. She met physics professor Bergen Davis, fell in love and they became engaged in 1906. The dean Laura Gill demanded her resignation, insisting she couldn't be both a physicist and a wife. Brooks protested, writing, "I think also it is a duty I owe to my profession and to my sex to show that a woman has a right to the practice of her profession and cannot be condemned to abandon it merely because she marries." The head of physics Margaret Maltby defended Brooks and her tremendous skills as a teacher and experimentalist, but the dean insisted she had to choose. Brooks ultimately broke off the engagement, and then nonetheless resigned due to the stress. She then took a step away from the scientific life.

In 1906 she spent her summer with the Fabian socialists at Summerbrook, the utopian commune established by Prestonia Mann Martin in the Adirondacks. She fell in with Marxist writer Maxim Gorky, his (then mistress, eventual second wife) Maria Andreyeva and their entourage and ended up, after a visit to Montreal, travelling with the couple by ship from New York to Naples and Capri! Eventually she seems to have bored of this and went to work with Marie Curie in Paris. Curie offered her a position for another year, but Rutherford, who was returning to England to work in Manchester offered her a fellowship. She accepted, but then suddenly withdrew. While in Montreal she had become reaquainted with her engineering tutor Frank Pitcher. He had been writing her ever since insisting that she should marry him as he could provide a stable future. She had been encouraged by her friends Mrs. Mary Rutherford (Ernest's wife) and Prestonia Martin (who both harboured strong traditional ideas about marriage) and she had ultimately gave in and accepted his proposal. He promptly took off on a mountain climbing tour of Europe, insisted on a religious ceremony despite her wishes and left her to return to Montreal and plan the wedding.

They had three children, but tragically lost one to spinal meningitis in childhood and a second to suicide while a student at McGill. Brooks, now Mrs. Frank Pitcher, never returned to research but took on the vocation of upper middle-class wife and mother until her premature death of leukimia at age 56, likely due to her exposure to Radon. She lived a quiet life after her remarkable 6 year science career, gardening and corresponding with those who had known her when she had been young and free. The social pressures and mores of the time, robbed physics of one of its bright lights.

Schematic electron shell diagram for Radon

I put a lot of thought into how to portray her. There is only a single, easily found, clear photo of her during her scientific career - her graduation photo. Showing her with a schematic diagram of Radon, like the photoillustration used by the Canadian Journal of Physics, seems an obvious choice. I considered that, and also, showing Thorium decaying to Radium to Radon, to allude to her discovery of Radon and chains of transmutations. However while these diagrams of atom are very graphic and make for an interesting image, they're misleading in a few ways. First, by showing the electron shells, we focus on chemistry and what's beyond the nucleus (and neither were what Brooks and colleagues were studying). Secondly, neither the nucleus nor electrons had been discovered during her time, so this is anarchronistic. I thought of showing nuclei undergoing alpha decay, and the daughters likewise decaying (more like my portrait of Lise Meitner), but this too whould be anachronistic; prior to 1906, no one had any idea there were protons and neutrons. So, I specifically showed what she discovered - atomic recoil - and depicted it simply, as it way understood at the time. I show Radium (just as a particle) spontaneously give off an alpha particle in one direction and emitting Radon in the other (what we now call the daughter nuclei). I chose this specific reaction to also reference her discovery of Radon itself. I considered showing the actual apparatus she used to investigate "Thorium emanations" since it's quite amazing that McGill has preserved these and posted photos online, but on its own it doesn't tell much of the story (unless you are already well-versed in the early history of research on radioactivity). I thought about showing our modern understanding of radioactive decay series (like the Uranium series above) but that also would have been anarchronistic. To allude to her third discovery, that these transmutations can occur in chains of reactions, at the bottom I have included a diagram that Rutherford published in 1905, labelled as the team at McGill then understood it: Radium gives off  an alpha particle and produces the "Emanation" (now Radon) which in turn gives off an alpha particle to produce "Radium A" (now Polonium), and a series of further transmutations by giving off alpha or beta or gamma particles and producing subsequent daughter nuclei, which at the time were simply labelled "Radium B" through "Radium F".  Her discoveries and work in collaboration with Rutherford, Thomson, Curie and others helped form the foundation for the entire field of nuclear physics and I hope my portrait can help bring her some of the attention she deserves as an important pioneer of the field.

* Chemists and biographers Marelene Rayner-Canham and Geoff Rayner-Canham point out the Matthew Effect (named by historian of science Robert Merton) where discoveries are often misattributed to a nearby more famous scientist. Consider the famous 'Rutherford Gold Foil Experiment', for instance, which was actually performed by his graduate students Geiger and Marsden, but by that time he had received the Nobel in chemistry "for his investigations into the disintegration of the elements, and the chemistry of radioactive substances." Historian of science Margaret Rossiter notes this Matthew Effect most commonly targets women and proposes the term Matilda Effect for the erasure of women from memory and the history of science.

Brooks could be poster child for the Matilda Effect.

** Famously pompous Thomson also disbelieved geological evidence for the age of the Earth, because thermal physics suggested the Earth might have cooled within a few thousand years (more in line with the Bible)... unless there was some other source of heat. There was: radioactivity. Remember, in fairness, at this time, neither the nucleus not the electron had been discovered. You may recall being taught Thomson's "raisin bun" model of the atom.

*** Unlike Pierre Curie, who threated to refuse the Nobel when he discovered that only he and Becquerel (and not Marie Curie) were to be credited with discovery of radioactivity, Otto Hahn did not insist that Meitner be included in his Nobel win (nor for that matter did he stand up for their assistant Fritz Straßmann). We now know that Meitner was nominated for a Nobel prize 47 times! 

It was mathematician Gösta Mittag-Leffler who alerted Pierre. An early ally to women in science, Gösta Mittag-Leffler also helped mathematician Sofia Kovalevski secure a position as a privat-docent at Stockholm University in Sweden in 1884. Kovalevski developed an intimate "romantic friendship" with his sister, actress, novelist, and playwright Duchess Anne-Charlotte Edgren-Leffler, with whom she lived collaborated on works of literature, for the remainder of her too short life.

References

"The Unstable Nucleus and its uses" American Institute of Physics exhibit on Marie Curie and the Science of Radioactivity - Radioactivity, accessed March 28, 2018

Emanations from Thorium and Radium Rutherford Museum at McGill, acessed March 28, 2018

Marelene Rayner-Canham and Geoff Rayner-Canham, "HARRIET BROOKS(1876-1933): CANADA'S FIRST WOMAN PHYSICIST" LA PHYSIQUE AU CANADA, 2005 - forms.cap.ca

Marelene F. Rayner-Canham and Geoffrey W. Rayner-Canham, comment on RUTHERFORD, THE “TRUE DISCOVERER OF RADON”, Bull. Hist. Chem., VOLUME 29, Number 2 (2004)

Elizabeth Shearly, Harriet Brooks, pioneering Canadian nuclear physicist, The Canadian Science Publishing blog, accessed March 28, 2018

John Geddes, Why Harriet Brooks Fits the Bill, Macleans, March 6, 2016

Ingrid Birker, Remembering Harriet Brooks: Canada’s first female nuclear physicist, McGill Reporter, March 8, 2011

"Harriet Brooks (1876-1933): Radioactivity" virtualmuseum.ca, accessed March 28, 2018

Brooks, Harriet (1876–1933) Women in World History: A Biographical Encyclopedia
COPYRIGHT 2002 Gale Research Inc.

Dale DeBakcsy, Wither: The Many Triumphs and Long Fall of Nuclear Physicist Harriet Brooks. (Women in Science 71), August 24, 2016

Harriet Brooks (Mrs. Frank Pitcher), Obituary, Nature 131, 865 (17 June 1933), doi:10.1038/131865a0

Inspired by Science on The Etsy Blog

The Etsy blog just posted Karen Brown's article featuring 5 Etsy artists who are inspired by science, including me!

Lise Meitner and Nuclear Fission Linocut History of Physics by minouette

 

“I think the idea that art and science are separate is unfounded,” says print maker Ele Willoughby of minouette. “It takes creativity to be a good scientist and experimentation to be a good artist.” In her Etsy shop, Ele explores art and science through a series of portraits of scientists inspired by the bi-monthly challenges of the Mad Scientists of Etsy team. “I love hearing from parents who want to inspire young children with portraits of scientific heroes or heroines,” she says.

There are some fabulous artists in that inspiring intersection of art and science, and several of my prints included.