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Magic number (physics)
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:
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 semiempirical 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 longestlived, 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 2126, 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. Additional recommended knowledge
Double magicNuclei 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 helium4 (^{4}He), oxygen16 (^{16}O),calcium40 (^{40}Ca), calcium48 (^{48}Ca), nickel48 (^{48}Ni), nickel56 (^{56}Ni), tin100 (^{100}Sn), tin132 (^{132}Sn) and lead208 (^{208}Pb). It is no accident that helium4 (^{4}He) and oxygen16 (^{16}O) are the second and third most abundant (and stable) nuclei in the universe^{[1]}. Both calcium48 (^{48}Ca) and nickel48 (^{48}Ni) are double magic because calcium48 has 20 protons and 28 neutrons while nickel48 has 28 protons and 20 neutrons. Calcium48 is very neutronrich for such a light element, but is made stable by being double magic. Similarly, nickel48, discovered in 1999, is the most protonrich isotope known^{[2]}. In December 2006 hassium270 (^{270}Hs) was discovered by an international team of scientists led by the Technical University of Munich having the unusually long halflife of 22 seconds. Hassium270 evidently forms part of an island of stability, and may even be double magic^{[3]}. DerivationMagic 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 spinorbit 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)(n^{2}+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 spinorbit 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)(n^{2}+5), giving series: 6, 14, 28, 50, 82, 126, 184 See also
References


This article is licensed under the GNU Free Documentation License. It uses material from the Wikipedia article "Magic_number_(physics)". A list of authors is available in Wikipedia. 