Magic number (physics)
Encyclopedia
In nuclear physics
, a magic number is a number of nucleon
s (either proton
s or neutron
s) such that they are arranged into complete shells within the atomic nucleus
. The seven most widely recognised 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 semi-empirical mass formula
and are hence more stable against nuclear decay.
The unusual stability of isotope
s having magic numbers means that transuranium element
s can be created with extremely large nuclei and yet not be subject to the extremely rapid radioactive decay
normally associated with high atomic number
s (as of 2007, the longest-lived, known isotope among all of the elements between 110 and 120 lasts only 12 min and the next 22 s). 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.
) numbers each 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
, oxygen-16 , calcium-40 , calcium-48
, nickel
-48 and lead
-208 .
Double-magic effects may allow existence of stable isotopes which otherwise would not have been expected. An example is calcium-40 , with 20 neutrons and 20 protons, which is the heaviest stable isotope made of the same number of protons and neutrons. Both calcium-48
and nickel
-48 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 like calcium-40, it is made stable by being double magic. Similarly, nickel-48, discovered in 1999, is the most proton-rich isotope known beyond helium-3.
Magic number shell effects are seen in ordinary abundances of elements: it is no accident that helium-4 is among the most abundant (and stable) nuclei in the universe and that lead-208 (208Pb) is the heaviest stable nuclide
.
Magic effects can keep unstable nuclides from decaying as rapidly as would otherwise be expected. For example, the nuclides tin
-100 and tin-132 are interesting examples of doubly magic isotopes of tin
that are unstable; however they represent endpoints beyond which stability drops off rapidly.
In December 2006 hassium
-270 , with 108 protons and 162 neutrons, 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.
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, magic numbers are the numbers of nucleons at which a shell is filled. For instance the magic number 8 occurs when 1s1/2, 1p3/2, 1p1/2 energy levels are filled as there is a large energy gap between the 1p1/2 and the next highest 1d5/2 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 gas
es helium, neon, argon, krypton, xenon, and radon. 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
.
Nuclear physics
Nuclear physics is the field of physics that studies the building blocks and interactions of atomic nuclei. The most commonly known applications of nuclear physics are nuclear power generation and nuclear weapons technology, but the research has provided application in many fields, including those...
, a magic number is a number of nucleon
Nucleon
In physics, a nucleon is a collective name for two particles: the neutron and the proton. These are the two constituents of the atomic nucleus. Until the 1960s, the nucleons were thought to be elementary particles...
s (either proton
Proton
The proton is a subatomic particle with the symbol or and a positive electric charge of 1 elementary charge. One or more protons are present in the nucleus of each atom, along with neutrons. The number of protons in each atom is its atomic number....
s or neutron
Neutron
The neutron is a subatomic hadron particle which has the symbol or , no net electric charge and a mass slightly larger than that of a proton. With the exception of hydrogen, nuclei of atoms consist of protons and neutrons, which are therefore collectively referred to as nucleons. The number of...
s) such that they are arranged into complete shells within the atomic nucleus
Atomic nucleus
The nucleus is the very dense region consisting of protons and neutrons at the center of an atom. It was discovered in 1911, as a result of Ernest Rutherford's interpretation of the famous 1909 Rutherford experiment performed by Hans Geiger and Ernest Marsden, under the direction of Rutherford. The...
. The seven most widely recognised magic numbers as of 2007 are
- 2, 8, 2020 (number)20 is the natural number following 19 and preceding 21. A group of twenty units may also be referred to as a score.-In mathematics:*20 is the basis for vigesimal number systems....
, 2828 (number)28 is the natural number following 27 and preceding 29.-In mathematics:It is a composite number, its proper divisors being 1, 2, 4, 7, and 14....
, 5050 (number)This article discusses the number fifty. For the year 50 CE, see 50. For other uses of 50, see 50 50 is the natural number following 49 and preceding 51.-In mathematics:...
, 8282 (number)82 is the natural number following 81 and preceding 83.-In mathematics:Eighty-two is the twenty-third biprime and the twelfth of the form...
, 126126 (number)126 is the natural number following 125 and preceding 127.-In mathematics:One hundred [and] twenty-six is a pentagonal pyramidal number, and a decagonal number as well as a pentatope number....
.
Atomic nuclei consisting of such a magic number of nucleons have a higher average binding energy
Binding energy
Binding energy is the mechanical energy required to disassemble a whole into separate parts. A bound system typically has a lower potential energy than its constituent parts; this is what keeps the system together—often this means that energy is released upon the creation of a bound state...
per nucleon
Nucleon
In physics, a nucleon is a collective name for two particles: the neutron and the proton. These are the two constituents of the atomic nucleus. Until the 1960s, the nucleons were thought to be elementary particles...
than one would expect based upon predictions such as the semi-empirical mass formula
Semi-empirical mass formula
In nuclear physics, the semi-empirical mass formula is used to approximate the mass and various other properties of an atomic nucleus...
and are hence more stable against nuclear decay.
The unusual stability of isotope
Isotope
Isotopes are variants of atoms of a particular chemical element, which have differing numbers of neutrons. Atoms of a particular element by definition must contain the same number of protons but may have a distinct number of neutrons which differs from atom to atom, without changing the designation...
s having magic numbers means that transuranium element
Transuranium element
In chemistry, transuranium elements are the chemical elements with atomic numbers greater than 92...
s can be created with extremely large nuclei and yet not be subject to the extremely rapid radioactive decay
Radioactive decay
Radioactive decay is the process by which an atomic nucleus of an unstable atom loses energy by emitting ionizing particles . The emission is spontaneous, in that the atom decays without any physical interaction with another particle from outside the atom...
normally associated with high atomic number
Atomic number
In chemistry and physics, the atomic number is the number of protons found in the nucleus of an atom and therefore identical to the charge number of the nucleus. It is conventionally represented by the symbol Z. The atomic number uniquely identifies a chemical element...
s (as of 2007, the longest-lived, known isotope among all of the elements between 110 and 120 lasts only 12 min and the next 22 s). Large isotopes with magic numbers of nucleons are said to exist in an island of stability
Island of stability
The island of stability in nuclear physics describes a set of as-yet undiscovered isotopes of transuranium elements which are theorized to be much more stable than others...
. 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.
Origin of the term
According to Steven A. Moszkowski (a student of Maria Goeppert-Mayer), the term "magic number" was coined by Eugene Wigner: "Wigner too believed in the liquid drop model, but he recognized, from the work of Maria Mayer, the very strong evidence for the closed shells. It seemed a little like magic to him, and that is how the words ‘Magic Numbers’ were coined.”Double magic
Nuclei which have neutron number and proton (atomicAtomic number
In chemistry and physics, the atomic number is the number of protons found in the nucleus of an atom and therefore identical to the charge number of the nucleus. It is conventionally represented by the symbol Z. The atomic number uniquely identifies a chemical element...
) numbers each 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
Helium-4
Helium-4 is a non-radioactive isotope of helium. It is by far the most abundant of the two naturally occurring isotopes of helium, making up about 99.99986% of the helium on earth. Its nucleus is the same as an alpha particle, consisting of two protons and two neutrons. Alpha decay of heavy...
, oxygen-16 , calcium-40 , calcium-48
Calcium-48
Calcium-48 is a rare isotope of calcium containing 20 protons and 28 neutrons. It makes up 0.187% of natural calcium by mole fraction. Although it is unusually neutron-rich for such a light nucleus, the only radioactive decay pathway open to it is the extremely rare process of double beta decay...
, nickel
Nickel
Nickel is a chemical element with the chemical symbol Ni and atomic number 28. It is a silvery-white lustrous metal with a slight golden tinge. Nickel belongs to the transition metals and is hard and ductile...
-48 and lead
Lead
Lead is a main-group element in the carbon group with the symbol Pb and atomic number 82. Lead is a soft, malleable poor metal. It is also counted as one of the heavy metals. Metallic lead has a bluish-white color after being freshly cut, but it soon tarnishes to a dull grayish color when exposed...
-208 .
Double-magic effects may allow existence of stable isotopes which otherwise would not have been expected. An example is calcium-40 , with 20 neutrons and 20 protons, which is the heaviest stable isotope made of the same number of protons and neutrons. Both calcium-48
Calcium-48
Calcium-48 is a rare isotope of calcium containing 20 protons and 28 neutrons. It makes up 0.187% of natural calcium by mole fraction. Although it is unusually neutron-rich for such a light nucleus, the only radioactive decay pathway open to it is the extremely rare process of double beta decay...
and nickel
Nickel
Nickel is a chemical element with the chemical symbol Ni and atomic number 28. It is a silvery-white lustrous metal with a slight golden tinge. Nickel belongs to the transition metals and is hard and ductile...
-48 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 like calcium-40, it is made stable by being double magic. Similarly, nickel-48, discovered in 1999, is the most proton-rich isotope known beyond helium-3.
Magic number shell effects are seen in ordinary abundances of elements: it is no accident that helium-4 is among the most abundant (and stable) nuclei in the universe and that lead-208 (208Pb) is the heaviest stable nuclide
Nuclide
A nuclide is an atomic species characterized by the specific constitution of its nucleus, i.e., by its number of protons Z, its number of neutrons N, and its nuclear energy state....
.
Magic effects can keep unstable nuclides from decaying as rapidly as would otherwise be expected. For example, the nuclides tin
Tin
Tin is a chemical element with the symbol Sn and atomic number 50. It is a main group metal in group 14 of the periodic table. Tin shows chemical similarity to both neighboring group 14 elements, germanium and lead and has two possible oxidation states, +2 and the slightly more stable +4...
-100 and tin-132 are interesting examples of doubly magic isotopes of tin
Isotopes of tin
Tin is the element with the greatest number of stable isotopes , which is probably related to the fact that 50 is a "magic number" of protons. 29 additional unstable isotopes are known, including the "doubly magic" tin-100 and tin-132 . The longest-lived radioisotope is 126Sn with a half-life...
that are unstable; however they represent endpoints beyond which stability drops off rapidly.
In December 2006 hassium
Hassium
Hassium is a synthetic element with the symbol Hs and atomic number 108. It is the heaviest member of the group 8 elements. The element was first observed in 1984...
-270 , with 108 protons and 162 neutrons, was discovered by an international team of scientists led by the Technical University of Munich
Technical University of Munich
The Technische Universität München is a research university with campuses in Munich, Garching, and Weihenstephan...
having the unusually long half-life
Half-life
Half-life, abbreviated t½, is the period of time it takes for the amount of a substance undergoing decay to decrease by half. The name was originally used to describe a characteristic of unstable atoms , but it may apply to any quantity which follows a set-rate decay.The original term, dating to...
of 22 seconds. Hassium-270 evidently forms part of an island of stability
Island of stability
The island of stability in nuclear physics describes a set of as-yet undiscovered isotopes of transuranium elements which are theorized to be much more stable than others...
, and may even be double magic.
Derivation
Magic numbers are typically obtained by empiricalEmpirical
The word empirical denotes information gained by means of observation or experimentation. Empirical data are data produced by an experiment or observation....
studies; however, if the form of the nuclear potential
Nuclear force
The nuclear force is the force between two or more nucleons. It is responsible for binding of protons and neutrons into atomic nuclei. The energy released causes the masses of nuclei to be less than the total mass of the protons and neutrons which form them...
is known then the Schrödinger equation
Schrödinger equation
The Schrödinger equation was formulated in 1926 by Austrian physicist Erwin Schrödinger. Used in physics , it is an equation that describes how the quantum state of a physical system changes in time....
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 1s1/2, 1p3/2, 1p1/2 energy levels are filled as there is a large energy gap between the 1p1/2 and the next highest 1d5/2 energy levels. The empirical values can be reproduced using the classical shell model with a strong spin-orbit interaction
Spin-orbit interaction
In quantum physics, the spin-orbit interaction is any interaction of a particle's spin with its motion. The first and best known example of this is that spin-orbit interaction causes shifts in an electron's atomic energy levels due to electromagnetic interaction between the electron's spin and...
.
The atomic analog to nuclear magic numbers are those numbers of electrons leading to discontinuities in the ionization energy. These occur for the noble gas
Noble gas
The noble gases are a group of chemical elements with very similar properties: under standard conditions, they are all odorless, colorless, monatomic gases, with very low chemical reactivity...
es helium, neon, argon, krypton, xenon, and radon. 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
Tetrahedron
In geometry, a tetrahedron is a polyhedron composed of four triangular faces, three of which meet at each vertex. A regular tetrahedron is one in which the four triangles are regular, or "equilateral", and is one of the Platonic solids...
.
See also
- Maria Göppert-Mayer
- Nuclear shell model
- SuperatomSuperatomSuperatoms are clusters of atoms that seem to exhibit some of the properties of elemental atoms.Sodium atoms, when cooled from vapor, naturally condense into clusters, preferentially containing a magic number of atoms . The first two of these can be recognized as the numbers of electrons needed to...
- SuperdeformationSuperdeformationIn nuclear physics a superdeformed nucleus is a nucleus that is very far from spherical, forming an ellipsoid with axes in ratios of approximately 2:1:1. Normal deformation is approximately 1.3:1:1. Only some nuclei can exist in superdeformed states....
External links
- OEIS : A018226 Magic numbers: atoms with one of these numbers of protons or neutrons in their nuclei are considered to be stable. see chapter 10 especially.