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 (either protons or neutrons) such that they are arranged into complete shells within the atomic nucleus. The seven known magic numbers as of 2007 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.

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 (the longest-lived isotope among all of the elements between 110 and 120 lasts only 3.6 seconds). Large isotopes with magic numbers of nucleons are said to exist in an island of stability.

1 Doubly magic
2 Derivation
3 References
4 See also
5 External links

[edit] Doubly magic

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. Examples of doubly magic isotopes include helium-4 (⁴He), oxygen-16 (¹⁶O), calcium-40 (⁴⁰Ca), calcium-48 (⁴⁸Ca), nickel-48 (⁴⁸Ni), nickel-56 (⁵⁶Ni), tin-100 (¹⁰⁰Sn), tin-132 (¹³²Sn) and lead-208 (²⁰⁸Pb). It is no accident that helium-4 (⁴He) and oxygen-16 (¹⁶O) are the second and third most abundant (and stable) elements in the universe^[1].

Both calcium-48 (⁴⁸Ca) and nickel-48 (⁴⁸Ni) are doubly magic because calcium-48 has 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 doubly magic. Similarly, nickel-48, discovered in 1999, is the most proton-rich isotope known^[2].

In December 2006 hassium-270 (²⁷⁰Hs) was discovered by an international team of scientists led by the Technical University of Munich having the unusually long half-life of 22 seconds. Hassium-270 evidently forms part of an island of stability, and may even be doubly magic^[3].

[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.

A non empirical derivation for all magic numbers, has been shown in the work published by Xavier Borg ^[4], where all magic numbers, including the theorized magic 184, are derived systematically from a hyper geometrical model based on two simplex stacked structures within the nucleus.

The lowest values 2, 8, and 20, closest to the stability line, agree with independent nucleon motion into a single particle potential, like a harmonic oscillator. Mathematically, the geometrical sequence of complete shell nucleon numbers for Z<=20 is given by the same equation that gives the total number of spheres of two symmetrical tetrahedral stacks forming an ideal 3D dipole:

For 3 ≥n≥ 1: Magic(n) = (n/3)(n²+3n+2), giving series: 2, 8, 20

The magic numbers 28, 50, 82, and 126, further away from the stability line, agree with those nuclei with a strong spin-orbit coupling (ref: Maria Mayer and Jensen). So, for Z>20, the total number of nucleons is given by the above equation, less a pair of two triangular layers which represent the binding energy (or nuclear anomalous missing mass). Thus:

For n>3: Magic(n) = (n/3)(n²+5), giving series: 28, 50, 82, 126, 184