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In Class 12 physics you learn that a nucleus is held together by binding energy, and that the binding energy per nucleon changes smoothly with mass number. Look closer, though, and some nuclei stand out as extra stable. Maria Goeppert Mayer explained why. This post covers her path, the magic numbers, a quick check you can do yourself, and what students can learn from how she worked.
Who was Maria Goeppert Mayer?
Her Nobel biography, written at the time of the award, says she was born on 28 June 1906 in Kattowitz, then in Germany, and that on her father's side she was the seventh straight generation of university professors. She grew up in Göttingen, enrolled at the university there in 1924 meaning to study mathematics, and turned to physics while quantum mechanics was young. She earned her doctorate in theoretical physics in 1930, with Max Born as her guide.
That year she married the American chemist Joseph Mayer and moved to Johns Hopkins University. The biography is blunt: in the Depression, "no university would think of employing the wife of a professor. But she kept working, just for the fun of doing physics." After Johns Hopkins and Columbia, she moved to Chicago in 1946, which the biography calls "the first place where she was not considered a nuisance, but greeted with open arms." Her facts page records her death on 20 February 1972 in San Diego.
What are the magic numbers?
In her Nobel lecture, The shell model, she lists them: 2, 8, 20, 28, 50, 82 and 126. A nucleus with a magic number of neutrons, or of protons, is unusually stable, and the numbers are the same for both.
She first noticed them while working with Edward Teller on the origin of the elements. Some nuclei were more abundant than any smooth theory predicted, and they shared a feature: 50 or 82 neutrons. The lecture gives more evidence. There are 6 stable nuclei with 50 neutrons and 7 with 82, whereas normally only 2 or 3 nuclei share a neutron number.
How did spin-orbit coupling explain them?
Simple models of nucleons moving in a potential well explained 2, 8 and 20, but the lecture says the magic numbers from 28 on "can definitely not be obtained" by extending that series. She describes the problem as a jigsaw puzzle with one piece missing.
The piece came from a question. Enrico Fermi, she recalls, asked as he was leaving her office: "Is there any indication of spin-orbit coupling?" Her answer: "Yes, of course and that will explain everything." She writes that Fermi was sceptical and left her "with my numerology". In ten minutes the magic numbers were explained. A week later, after she had written up the consequences, he was convinced and taught the idea in his nuclear physics class.
The basic assumption, in her words, is "a strong interaction between spin and orbital angular momentum", which gives the level with total angular momentum j = l + 1/2 a considerably lower energy than j = l − 1/2. The split grows with l, so big gaps open where a high-l level first appears. Her lecture works through 28 as the example:
The simpler oscillator shells close at 20.
The next level is 1f, with l = 3. It splits into j = 7/2 and j = 5/2, with 7/2 lower.
A level with angular momentum j holds 2j + 1 nucleons, so the 7/2 level holds 2 × 7/2 + 1 = 8.
The large gap above it therefore falls at 20 + 8 = 28.
She published it in On Closed Shells in Nuclei. II in Physical Review in 1949. Otto Haxel, Hans Jensen and Hans Suess proposed the same explanation independently, in On the "Magic Numbers" in Nuclear Structure, the same year.
Try it: which nuclei are doubly magic?
A nucleus is doubly magic when both its proton number Z and neutron number N = A − Z are magic. Check these:
Oxygen-16: Z = 8, N = 16 − 8 = 8. Both magic.
Calcium-40: Z = 20, N = 40 − 20 = 20. Both magic.
Calcium-48: Z = 20, N = 48 − 20 = 28. Both magic.
Lead-208: Z = 82, N = 208 − 82 = 126. Both magic.
Iron-56: Z = 26, N = 56 − 26 = 30. Neither is magic.
The common mistake is to test the mass number A instead of Z and N separately. Lead-208 is doubly magic because 82 and 126 are each magic, not because 208 is.
Recognition
According to the 1963 prize summary, she and J. Hans D. Jensen shared one half of the Nobel Prize in Physics "for their discoveries concerning nuclear shell structure", and Eugene Wigner received the other half for separate work. The Nobel Prize's list of women laureates shows only Marie Curie, in 1903, had won the physics prize before her. Read more about that first win in our post on Marie Curie's two Nobel Prizes.
What can a student take from her way of working?
Live with the data. She could answer Fermi's question at once only because, as she put it, she had lived with the data so long. Familiarity with many solved problems lets you spot patterns quickly.
Plot things differently. The lecture says the pattern appears when you plot against neutron or proton number rather than mass number. Changing the axis can reveal what a table hides.
Expect false starts. She writes that she had made many before the one that worked.
Keep working without recognition. For years she worked without a proper post. Steady work counted later.
To practise the binding energy calculations behind all this, use our guide to mass defect and binding energy problems. When a nuclear numerical goes wrong at the set-up stage, the Fermi Problem Library has worked problems with a step-by-step solver that explains the reasoning rather than printing an answer.
Next step: check whether tin-132 and nickel-56 are doubly magic, using Z = 50 and Z = 28.
