Magic number (physics)

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A graph of isotope stability, with some of the magic numbers. Table isotopes en.svg
A graph of isotope stability, with some of the magic numbers.

In nuclear physics, a magic number is a number of nucleons (either protons or neutrons, separately) such that they are arranged into complete shells within the atomic 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 are 2, 8, 20, 28, 50, 82, and 126.

Contents

For protons, this corresponds to the elements helium, oxygen, calcium, nickel, tin, lead, and the hypothetical unbihexium, 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 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 could theoretically be created with extremely large nuclei and yet not be subject to the extremely rapid radioactive decay normally associated with high atomic numbers. 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. [1] [2] [3]

The difference between known binding energies of isotopes and the binding energy as predicted from the semi-empirical mass formula. Distinct sharp peaks in the contours appear only at magic numbers. Semi-empirical mass formula discrepancy.png
The difference between known binding energies of isotopes and the binding energy as predicted from the semi-empirical mass formula. Distinct sharp peaks in the contours appear only at magic numbers.

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 (see Binomial 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]

History and etymology

Maria Goeppert Mayer Maria Goeppert-Mayer portrait.jpg
Maria Goeppert Mayer

Upon working on the Manhattan Project, the German physicist Maria 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 physicist Eugene 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 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." [11]

These magic numbers were the bedrock of the nuclear shell model, which Mayer developed in the following years together with Hans Jensen and culminated in their shared 1963 Nobel Prize in Physics. [12]

Doubly magic

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 are helium-4, helium-10, oxygen-16, calcium-40, calcium-48, nickel-48, nickel-56, nickel-78, tin-100, tin-132, and lead-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 inefficient double 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 the age 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 a half-life of just 260(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 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 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, although oxygen-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 heaviest stable nuclide (at least by known experimental observations). Alpha decay (the emission of a 4He 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 over neutron emission, proton emission or any other type of cluster decay. The stability of 4He also leads to the absence of stable isobars 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 magic isotopes 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 from tritium 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 the Technical University of Munich, having a half-life of 9 seconds. [18] Hassium-270 evidently forms part of an island of stability, and may even be doubly magic due to the deformed (American football- or rugby ball-like) shape of this nucleus. [19] [20]

Although Z = 92 and N = 164 are not magic numbers, the undiscovered neutron-rich nucleus uranium-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 the nuclear potential. [21]

Derivation

Magic numbers are typically obtained by empirical studies; 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 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 of electrons leading to discontinuities in the ionization energy. These occur for the noble gases helium, neon, argon, krypton, xenon, radon and oganesson. 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. Hence copernicium (112) and flerovium (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 the fractional 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]

See also

Related Research Articles

<span class="mw-page-title-main">Atom</span> Smallest unit of a chemical element

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Hassium is a synthetic chemical element; it has symbol Hs and atomic number 108. It is highly radioactive: its most stable known isotopes have half-lives of approximately ten seconds. One of its isotopes, 270Hs, has magic numbers of protons and neutrons for deformed nuclei, giving it greater stability against spontaneous fission. Hassium is a superheavy element; it has been produced in a laboratory in very small quantities by fusing heavy nuclei with lighter ones. Natural occurrences of the element have been hypothesised but never found.

<span class="mw-page-title-main">Nuclear physics</span> Field of physics that studies atomic nuclei

Nuclear physics is the field of physics that studies atomic nuclei and their constituents and interactions, in addition to the study of other forms of nuclear matter.

<span class="mw-page-title-main">Island of stability</span> Predicted set of isotopes of relatively more stable superheavy elements

In nuclear physics, the island of stability is a predicted set of isotopes of superheavy elements that may have considerably longer half-lives than known isotopes of these elements. It is predicted to appear as an "island" in the chart of nuclides, separated from known stable and long-lived primordial radionuclides. Its theoretical existence is attributed to stabilizing effects of predicted "magic numbers" of protons and neutrons in the superheavy mass region.

Moscovium is a synthetic chemical element; it has symbol Mc and atomic number 115. It was first synthesized in 2003 by a joint team of Russian and American scientists at the Joint Institute for Nuclear Research (JINR) in Dubna, Russia. In December 2015, it was recognized as one of four new elements by the Joint Working Party of international scientific bodies IUPAC and IUPAP. On 28 November 2016, it was officially named after the Moscow Oblast, in which the JINR is situated.

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<span class="mw-page-title-main">Nuclide</span> Atomic species

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Unbibium, also known as element 122 or eka-thorium, is a hypothetical chemical element; it has placeholder symbol Ubb and atomic number 122. Unbibium and Ubb are the temporary systematic IUPAC name and symbol respectively, which are used until the element is discovered, confirmed, and a permanent name is decided upon. In the periodic table of the elements, it is expected to follow unbiunium as the second element of the superactinides and the fourth element of the 8th period. Similarly to unbiunium, it is expected to fall within the range of the island of stability, potentially conferring additional stability on some isotopes, especially 306Ubb which is expected to have a magic number of neutrons (184).

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, 61
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, 62
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 and 64
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 with a half-life of 76,000 years, 63
Ni
 with a half-life of 100.1 years, and 56
Ni
 with a half-life of 6.077 days. All of the remaining radioactive isotopes have half-lives that are less than 60 hours and the majority of these have half-lives that are less than 30 seconds. This element also has 8 meta states.

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References

  1. 1 2 Kratz, J. V. (5 September 2011). The Impact of Superheavy Elements on the Chemical and Physical Sciences (PDF). 4th International Conference on the Chemistry and Physics of the Transactinide Elements. Retrieved 27 August 2013.
  2. ↑ "Nuclear scientists eye future landfall on a second 'island of stability'".
  3. ↑ Grumann, Jens; Mosel, Ulrich; Fink, Bernd; Greiner, Walter (1969). "Investigation of the stability of superheavy nuclei aroundZ=114 andZ=164". Zeitschrift für Physik. 228 (5): 371–386. Bibcode:1969ZPhy..228..371G. doi:10.1007/BF01406719. S2CID   120251297.
  4. ↑ "Nuclear scientists eye future landfall on a second 'island of stability'".
  5. ↑ Grumann, Jens; Mosel, Ulrich; Fink, Bernd; Greiner, Walter (1969). "Investigation of the stability of superheavy nuclei aroundZ=114 andZ=164". Zeitschrift für Physik. 228 (5): 371–386. Bibcode:1969ZPhy..228..371G. doi:10.1007/BF01406719. S2CID   120251297.
  6. ↑ Koura, H. (2011). Decay modes and a limit of existence of nuclei in the superheavy mass region (PDF). 4th International Conference on the Chemistry and Physics of the Transactinide Elements. Retrieved 18 November 2018.
  7. ↑ Out of the shadows : contributions of twentieth-century women to physics. Byers, Nina. Cambridge: Cambridge Univ. Pr. 2006. ISBN   0-521-82197-5. OCLC   255313795.{{cite book}}: CS1 maint: others (link)
  8. ↑ Mayer, Maria G. (1948-08-01). "On Closed Shells in Nuclei". Physical Review. 74 (3): 235–239. Bibcode:1948PhRv...74..235M. doi:10.1103/physrev.74.235. ISSN   0031-899X.
  9. ↑ Wigner, E. (1937-01-15). "On the Consequences of the Symmetry of the Nuclear Hamiltonian on the Spectroscopy of Nuclei". Physical Review. 51 (2): 106–119. Bibcode:1937PhRv...51..106W. doi:10.1103/PhysRev.51.106.
  10. ↑ Mayer, Maria Goeppert (1949-06-15). "On Closed Shells in Nuclei. II". Physical Review. 75 (12): 1969–1970. Bibcode:1949PhRv...75.1969M. doi:10.1103/PhysRev.75.1969.
  11. ↑ Audi, Georges (2006). "The history of nuclidic masses and of their evaluation". International Journal of Mass Spectrometry. 251 (2–3): 85–94. arXiv: physics/0602050 . Bibcode:2006IJMSp.251...85A. doi:10.1016/j.ijms.2006.01.048. S2CID   13236732.
  12. ↑ "The Nobel Prize in Physics 1963". NobelPrize.org. Retrieved 2020-06-27.
  13. ↑ "What is Stable Nuclei - Unstable Nuclei - Definition". Periodic Table. 2019-05-22. Retrieved 2019-12-22.
  14. ↑ Kondo, Y.; Achouri, N. L.; Falou, H. Al; Atar, L.; Aumann, T.; Baba, H.; Boretzky, K.; Caesar, C.; Calvet, D.; Chae, H.; Chiga, N.; Corsi, A.; Delaunay, F.; Delbart, A.; Deshayes, Q. (2023-08-31). "First observation of 28O". Nature. 620 (7976): 965–970. doi:10.1038/s41586-023-06352-6. ISSN   0028-0836. PMC   10630140 . PMID   37648757.
  15. ↑ Nave, C. R. "The Most Tightly Bound Nuclei". HyperPhysics.
  16. ↑ W., P. (October 23, 1999). "Twice-magic metal makes its debut - isotope of nickel". Science News. Archived from the original on May 24, 2012. Retrieved 2006-09-29.
  17. ↑ "Tests confirm nickel-78 is a 'doubly magic' isotope". Phys.org. September 5, 2014. Retrieved 2014-09-09.
  18. ↑ Audi, G.; Kondev, F. G.; Wang, M.; Huang, W. J.; Naimi, S. (2017). "The NUBASE2016 evaluation of nuclear properties" (PDF). Chinese Physics C. 41 (3): 030001–134. Bibcode:2017ChPhC..41c0001A. doi:10.1088/1674-1137/41/3/030001.
  19. ↑ Mason Inman (2006-12-14). "A Nuclear Magic Trick". Physical Review Focus. Vol. 18. Retrieved 2006-12-25.
  20. ↑ Dvorak, J.; Brüchle, W.; Chelnokov, M.; Dressler, R.; Düllmann, Ch. E.; Eberhardt, K.; Gorshkov, V.; Jäger, E.; Krücken, R.; Kuznetsov, A.; Nagame, Y.; Nebel, F.; Novackova, Z.; Qin, Z.; Schädel, M.; Schausten, B.; Schimpf, E.; Semchenkov, A.; Thörle, P.; Türler, A.; Wegrzecki, M.; Wierczinski, B.; Yakushev, A.; Yeremin, A. (2006). "Doubly Magic Nucleus 108270Hs162". Physical Review Letters. 97 (24): 242501. Bibcode:2006PhRvL..97x2501D. doi:10.1103/PhysRevLett.97.242501. PMID   17280272.
  21. ↑ Koura, H.; Chiba, S. (2013). "Single-Particle Levels of Spherical Nuclei in the Superheavy and Extremely Superheavy Mass Region". Journal of the Physical Society of Japan. 82 (1): 014201. Bibcode:2013JPSJ...82a4201K. doi:10.7566/JPSJ.82.014201.
  22. ↑ Herrmann, Richard (2010). "Higher dimensional mixed fractional rotation groups as a basis for dynamic symmetries generating the spectrum of the deformed Nilsson-oscillator". Physica A . 389 (4): 693–704. arXiv: 0806.2300 . Bibcode:2010PhyA..389..693H. doi:10.1016/j.physa.2009.11.016.
  23. ↑ Herrmann, Richard (2010). "Fractional phase transition in medium size metal clusters and some remarks on magic numbers in gravitationally and weakly bound clusters". Physica A. 389 (16): 3307–3315. arXiv: 0907.1953 . Bibcode:2010PhyA..389.3307H. doi:10.1016/j.physa.2010.03.033. S2CID   50477979.
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