In nuclear physics, a magic number is a number of nucleons (either protons or neutrons, separately) such that they are arranged into complete shells within the atomic nucleus. As a result, atomic nuclei with a "magic" number of protons or neutrons are much more stable than other nuclei. The seven most widely recognized magic numbers as of 2019 are 2, 8, 20, 28, 50, 82, and 126. For protons, this corresponds to the elements helium, oxygen, calcium, nickel, tin, lead, and the hypothetical unbihexium, although 126 is so far only known to be a magic number for neutrons. Atomic nuclei consisting of such a magic number of nucleons have a higher average binding energy per nucleon than one would expect based upon predictions such as the semi-empirical mass formula and are hence more stable against nuclear decay. The unusual stability of isotopes having magic numbers means that transuranium elements could theoretically be created with extremely large nuclei and yet not be subject to the extremely rapid radioactive decay normally associated with high atomic numbers. Large isotopes with magic numbers of nucleons are said to exist in an island of stability. Unlike the magic numbers 2–126, which are realized in spherical nuclei, theoretical calculations predict that nuclei in the island of stability are deformed.
Before this was realized, higher magic numbers, such as 184, 258, 350, and 462, were predicted based on simple calculations that assumed spherical shapes: these are generated by the formula 2 ( ( n 1 ) + ( n 2 ) + ( n 3 ) ) {\displaystyle 2({\tbinom {n}{1}}+{\tbinom {n}{2}}+{\tbinom {n}{3}})} (see Binomial coefficient). It is now believed that the sequence of spherical magic numbers cannot be extended in this way. Further predicted magic numbers are 114, 122, 124, and 164 for protons as well as 184, 196, 236, and 318 for neutrons. However, more modern calculations predict 228 and 308 for neutrons, along with 184 and 196.
History and etymology
While working on the Manhattan Project, the German physicist Maria Goeppert Mayer became interested in the properties of nuclear fission products, such as decay energies and half-lives. In 1948, she published a body of experimental evidence for the occurrence of closed nuclear shells for nuclei with 50 or 82 protons or 50, 82, and 126 neutrons. It had already been known that nuclei with 20 protons or neutrons were stable: that was evidenced by calculations by Hungarian-American physicist Eugene Wigner, one of her colleagues in the Manhattan Project. Two years later, in 1950, a new publication followed in which she attributed the shell closures at the magic numbers to spin-orbit coupling. According to Steven Moszkowski, a student of Goeppert Mayer, the term "magic number" was coined by Wigner: "Wigner too believed in the liquid drop model, but he recognized, from the work of Maria Mayer, the very strong evidence for the closed shells. It seemed a little like magic to him, and that is how the words 'Magic Numbers' were coined." These magic numbers were the bedrock of the nuclear shell model, which Mayer developed in the following years together with Hans Jensen and culminated in their shared 1963 Nobel Prize in Physics.
Doubly magic
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