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In nuclear physics, a magic number is a number of nucleons (either protons or neutrons) such that they are arranged into complete shells within the atomic nucleus. The seven most widely recognised magic numbers as of 2007 are
2, 8, 20, 28, 50, 82, 126.

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 can be created with extremely large nuclei and yet not be subject to the extremely rapid radioactive decay normally associated with high atomic numbers (as of 2007, the longest-lived, known isotope among all of the elements between 110 and 120 lasts only 12 min., next 22 sec.). 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, were predicted based on simple calculations that assumed spherical shapes. It is now believed that the sequence of spherical magic numbers cannot be extended in this way.

Double magic

Nuclei which have both neutron number and proton (atomic) number equal to one of the magic numbers are called "double magic", and are especially stable against decay. Examples of double magic isotopes include helium-4 (4He), oxygen-16 (16O), calcium-40 (40Ca), calcium-48 (48Ca), nickel-48 (48Ni) and lead-208 (208Pb). Tin-100 (100Sn) and tin-132 (132Sn) are doubly-magic isotopes of tin that are unstable; however they represent endpoints beyond which stability drops off rapidly. It is no accident that helium-4 (4He) is among the most abundant (and stable) nuclei in the universe and that lead-208 (208Pb) is the heaviest stable nuclide.

Both calcium-48 (48Ca) and nickel-48 (48Ni) are double magic because calcium-48 has 20 protons and 28 neutrons while nickel-48 has 28 protons and 20 neutrons. Calcium-48 is very neutron-rich for such a light element, but is made stable by being double magic. Similarly, nickel-48, discovered in 1999, is the most proton-rich isotope known beyond helium-3.

In December 2006 hassium-270 (270Hs) was discovered by an international team of scientists led by the Technical University of Munichmarker having the unusually long half-life of 22 seconds. Hassium-270 evidently forms part of an island of stability, and may even be double magic.


Magic numbers are typically obtained by empirical studies; however, if the form of the nuclear potential is known then the Schrödinger equation can be solved for the motion of nucleons and energy levels determined. Nuclear shells are said to occur when the separation between energy levels is significantly greater than the local mean separation.

In the shell model for the nucleus, magic numbers are the numbers of nucleons at which a shell is filled. For instance the magic number 8 occurs when 1s1/2, 1p3/2, 1p1/2 energy levels are filled as there is a large energy gap between the 1p1/2 and the next highest 1d5/2 energy levels. The empirical values can be reproduced using the classical shell model with a strong spin-orbit interaction.

The atomic analog to nuclear magic numbers are those numbers of electrons leading to discontinuities in the ionization energy. These occur for the noble gases, and hence, the "atomic magic numbers" are 2, 10, 18, 36, 54, and 86.

In 2007, Jozsef Garai from Florida International University proposed a mathematical formula describing the periodicity of the nucleus in the periodic system based on the tetrahedron.

In 2009, Jean-claude Perez proposed in a book a very simple numerical formula computing the number of elements within every period then, finally, modelling the periodic system whole structure.

c(p) = 2 [ Int ( (p+2) / 2 ) ]**2

Where p is the period, c(p) is the number of elements within the period p, and Int(x) is the whole integer part of the decimal value x.

See also


  1. Hyperphysics
  2. Jean-claude Perez, CODEX BIOGENESIS , (2009) Marco Pietteur publishing (Resurgence collection) Embourg Belgium, ISBN 2874340448. pages 47-70 (in french)

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