Innuclear physics, amagic number is a number ofnucleons (eitherprotons orneutrons, separately) such that they are arranged into completeshells within theatomic nucleus. As a result, atomic nuclei with a "magic" number of protons or neutrons are much more stable than other nuclei. The seven most widely recognized magic numbers as of 2019 are2, 8, 20, 28, 50, 82, and126.
For protons, this corresponds to the elementshelium,oxygen,calcium,nickel,tin,lead, and the hypotheticalunbihexium, although 126 is so far only known to be a magic number for neutrons. Atomic nuclei consisting of such a magic number of nucleons have a higher averagebinding energy pernucleon than one would expect based upon predictions such as thesemi-empirical mass formula and are hence more stable against nuclear decay.
The unusual stability ofisotopes having magic numbers means thattransuranium elements could theoretically be created with extremely large nuclei and yet not be subject to the extremely rapidradioactive decay normally associated with highatomic numbers. Large isotopes with magic numbers of nucleons are said to exist in anisland 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.[1][2][3]
Before this was realized, higher magic numbers, such as 184, 258, 350, and 462, were predicted based on simple calculations that assumed spherical shapes: these are generated by the formula (seeBinomial coefficient). It is now believed that the sequence of spherical magic numbers cannot be extended in this way. Further predicted magic numbers are 114, 122, 124, and 164 for protons as well as 184, 196, 236, and 318 for neutrons.[1][4][5] However, more modern calculations predict 228 and 308 for neutrons, along with 184 and 196.[6]
While working on theManhattan Project, the German physicistMaria Goeppert Mayer became interested in the properties of nuclear fission products, such as decay energies and half-lives.[7] In 1948, she published a body of experimental evidence for the occurrence of closed nuclear shells for nuclei with 50 or 82 protons or 50, 82, and 126 neutrons.[8]
It had already been known that nuclei with 20 protons or neutrons were stable: that was evidenced by calculations by Hungarian-American physicistEugene Wigner, one of her colleagues in the Manhattan Project.[9] Two years later, in 1950, a new publication followed in which she attributed the shell closures at the magic numbers to spin-orbit coupling.[10] According to Steven Moszkowski, a student of Goeppert Mayer, the term "magic number" was coined by Wigner: "Wigner too believed in theliquid 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."[11]
These magic numbers were the bedrock of thenuclear shell model, which Mayer developed in the following years together withHans Jensen and culminated in their shared1963 Nobel Prize in Physics.[12]
Nuclei which have neutron numbers and proton (atomic) numbers both equal to one of the magic numbers are called "doubly magic", and are generally very stable against decay.[13] The known doubly magic isotopes arehelium-4,helium-10,oxygen-16,calcium-40,calcium-48,nickel-48, nickel-56, nickel-78,tin-100, tin-132, andlead-208. While only helium-4, oxygen-16, calcium-40, and lead-208 are completely stable, calcium-48 is extremely long-lived and therefore found naturally, disintegrating only by a very inefficientdouble beta minus decay process. Double beta decay in general is so rare that several nuclides exist which are predicted to decay by this mechanism but in which no such decay has yet been observed. Even in nuclides whose double beta decay has been confirmed through observations, half lives usually exceed theage of the universe by orders of magnitude, and emitted beta or gamma radiation is for virtually all practical purposes irrelevant. On the other hand, helium-10 is extremely unstable, and has ahalf-life of just260(40) yoctoseconds (2.6(4)×10−22 s).
Doubly magic effects may allow the existence of stable isotopes which otherwise would not have been expected. An example iscalcium-40, with 20 neutrons and 20 protons, which is the heaviest stable isotope made of the same number of protons and neutrons. Bothcalcium-48 andnickel-48 are doubly 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 relatively light element, but like calcium-40, it is stabilized by being doubly magic. As an exception, althoughoxygen-28 has 8 protons and 20 neutrons, it is unbound with respect to four-neutron decay and appears to lack closed neutron shells, so it is not regarded as doubly magic.[14]
Magic number shell effects are seen in ordinary abundances of elements: helium-4 is among the most abundant (and stable) nuclei in the universe[15] and lead-208 is the heavieststablenuclide (at least by known experimental observations).Alpha decay (the emission of a4He nucleus – also known as an alpha particle – by a heavy element undergoing radioactive decay) is common in part due to the extraordinary stability of helium-4, which makes this type of decay energetically favored in most heavy nuclei overneutron emission,proton emission or any other type ofcluster decay. The stability of4He also leads to the absence of stableisobars of mass number 5 and 8; indeed, all nuclides of those mass numbers decay within fractions of a second to produce alpha particles.
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 examples of doubly magicisotopes of tin that are unstable, and represent endpoints beyond which stability drops off rapidly. Nickel-48, discovered in 1999, is the most proton-rich doubly magic nuclide known.[16] At the other extreme, nickel-78 is also doubly magic, with 28 protons and 50 neutrons, a ratio observed only in much heavier elements, apart fromtritium with one proton and two neutrons (78Ni: 28/50 = 0.56;238U: 92/146 = 0.63).[17]
In December 2006,hassium-270, with 108 protons and 162 neutrons, was discovered by an international team of scientists led by theTechnical University of Munich, having ahalf-life of 9 seconds.[18] Hassium-270 evidently forms part of anisland of stability, and may even be doubly magic due to the deformed (American football- orrugby ball-like) shape of this nucleus.[19][20]
AlthoughZ = 92 andN = 164 are not magic numbers, the undiscovered neutron-rich nucleusuranium-256 may be doubly magic and spherical due to the difference in size between low- and high-angular momentum orbitals, which alters the shape of thenuclear potential.[21]
Magic numbers are typically obtained byempirical studies; if the form of thenuclear potential is known, then theSchrödinger 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 theshell 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 the 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 atomic analog to nuclear magic numbers are those numbers ofelectrons leading to discontinuities in theionization energy. These occur for thenoble gaseshelium,neon,argon,krypton,xenon,radon andoganesson. Hence, the "atomic magic numbers" are 2, 10, 18, 36, 54, 86 and 118. As with the nuclear magic numbers, these are expected to be changed in the superheavy region due to spin/orbit-coupling effects affecting subshell energy levels. Hencecopernicium (112) andflerovium (114) are expected to be more inert than oganesson (118), and the next noble gas after these is expected to occur at element 172 rather than 168 (which would continue the pattern).
In 2010, an alternative explanation of magic numbers was given in terms of symmetry considerations. Based on thefractional extension of the standard rotation group, the ground state properties (including the magic numbers) for metallic clusters and nuclei were simultaneously determined analytically. A specific potential term is not necessary in this model.[22][23]
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