Magic number (physics)

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)

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 11 seconds). 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.

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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), 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) nuclei in the universe^[1].

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^[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 double magic^[3].

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.

In the shell model for the nucleus, 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. A highly simplified version of this is the shell model with a deformed harmonic oscillator potential and spin-orbit interaction.

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 > 1: Magic(n) = (n/3)(n²+5), giving series: 6, 14, 28, 50, 82, 126, 184

See also

• Island of stability
• Isotopes
• Maria Göppert-Mayer
• Nuclear shell model
• Superatom
• Superdeformation
• Semi-empirical mass formula

References