Magic number (physics)

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For other uses of the term, see magic number

In nuclear physics, a magic number is a number of nucleons such that they are arranged into complete shells within the atomic nucleus. The seven known magic numbers as of 2006 are:

2, 8, 20, 28, 50, 82, 126 (sequence A018226 in OEIS)

It is also theorized that 184 is a magic number.

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. Nuclei which have both neutron number and proton (atomic) number equal to one of the magic numbers are called doubly magic, and are especially stable against decay. For example, Lead-208 is especially stable as it has both 82 protons and 126 neutrons. And calcium-48 has 20 protons and 28 neutrons, so is very stable although it is very neutron-rich for such a light element.

[edit] Derivation

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.

Magical numbers are the numbers of nucleons at which a shell is filled. For instance the magic number 8 occurs when 1s_1/2, 1p_3/2, 1p_1/2 energy levels are filled as there is a large energy gap between the 1p_1/2 and the next highest 1d_5/2 energy levels.