magic numbers

magic numbers [′maj·ik ′nəm·bərz]

(nuclear physics)

The integers 8, 20, 28, 50, 82, 126; nuclei in which the number of protons, neutrons, or both is magic have a stability and binding energy which is greater than average, and have other special properties.

(physical chemistry)

Numbers of atoms or molecules for which certain atom or molecular clusters have an unusually high abundance.

---

Magic numbers

The number of neutrons or protons in nuclei which are required to fill major quantum shells. They occur at particle numbers 2, 8, 20, 50, and 82.

In atoms, the electrons that orbit the nucleus fill quantum electron shells at atomic numbers of 2 (helium), 10 (neon), 18 (argon), 36 (krypton), and 54 (xenon). These elements are chemically inert and difficult to ionize because the energies of orbits are grouped in bunches or shells with large gaps between them. In nuclei, an analogous behavior is found; quantum orbits completely filled with neutrons or protons result in extra stability. The neutrons and protons fill their quantum states independently, so that both full neutron and full proton shells can occur as magic nuclei. In a few cases, for example oxygen-16 (¹⁶₈O₈) and calcium-40 (⁴⁰₂₀Ca₂₀), doubly magic nuclei have full neutron and proton shells. Between the major shell gaps, smaller subshell gaps cause some extra stabilization and semimagic behavior is found at particle numbers 14, 28, 40, and 64. See Atomic structure and spectra, Electron configuration

In very heavy nuclei the Coulomb repulsion between the protons results in a different sequence of states for neutrons and protons and different major shell gaps. For neutrons the magic sequence continues at N = 126; the next shell gap is predicted at N = 184. For protons the next major shell gap is anticipated at Z = 114. The latter shell gaps lie beyond the heaviest nuclei known, but calculations indicate that the extra stability gained by producing nuclei with these particle numbers may result in an island of long-lived superheavy nuclei.

The closing of nuclear quantum shells has many observable consequences. The nuclei are more tightly bound than average, and the extra stability leads to anomalously high abundances of magic nuclei in nature. The full shells require unusually high energies to remove the least bound neutron or proton, and the probability of capturing extra particles is lower than expected. Furthermore, the full shells are spherically symmetric, and the nuclei have very small electric quadrupole moments. Many of these properties were known before the nuclear shell model was developed to account for quantum-level ordering and gaps between major shells. The different shell closures for atomic and nuclear systems reflect the differences between the Coulomb force that binds electrons to nuclei and the strong force that holds the nucleus together. An important component of the strong force in nuclei is the spin-orbit term, which makes the energy of a state strongly dependent on the relative orientation of spin and orbital angular momentum. See Angular momentum, Isotope, Nuclear moments, Nuclear structure, Strong nuclear interactions