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 (
^{4}He),
oxygen-16 (
^{16}O),
calcium-40 (
^{40}Ca),
calcium-48 (
^{48}Ca),
nickel-48 (
^{48}Ni) and
lead-208 (
^{208}Pb).
Tin-100
(
^{100}Sn) and
tin-132
(
^{132}Sn) 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
(
^{4}He) is among the most abundant (and stable) nuclei in
the universe and that lead-208 (
^{208}Pb) is the heaviest
stable
nuclide.
Both
calcium-48 (
^{48}Ca) and
nickel-48 (
^{48}Ni) 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 (^{270}Hs)
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.
Derivation
Magic numbers are typically obtained by
empirical studies; however, if the form of the
nuclear potential is known then the
SchrĂ¶dinger 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
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. 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
References
- Hyperphysics
- Jean-claude Perez, CODEX BIOGENESIS , (2009) Marco Pietteur
publishing (Resurgence collection) Embourg Belgium, ISBN
2874340448. pages 47-70 (in french)
External links