Magic number (physics)

Last updated
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(sequence A018226 in the OEIS ).

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 (sequence A033547 in the OEIS ), 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

Atoms are the basic particles of the chemical elements. An atom consists of a nucleus of protons and generally neutrons, surrounded by an electromagnetically bound swarm of electrons. The chemical elements are distinguished from each other by the number of protons that are in their atoms. For example, any atom that contains 11 protons is sodium, and any atom that contains 29 protons is copper. Atoms with the same number of protons but a different number of neutrons are called isotopes of the same element.

<span class="mw-page-title-main">Hassium</span> Chemical element, symbol Hs and atomic number 108

Hassium is a chemical element; it has symbol Hs and atomic number 108. Hassium is highly radioactive: its most stable known isotopes have half-lives of approximately ten seconds. One of its isotopes, 270Hs, has magic numbers of both protons and neutrons for deformed nuclei, which gives it greater stability against spontaneous fission. Hassium is a superheavy element; it has been produced in a laboratory only 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">Stable nuclide</span> Nuclide that does not undergo radioactive decay

Stable nuclides are nuclides that are not radioactive and so do not spontaneously undergo radioactive decay. When such nuclides are referred to in relation to specific elements, they are usually termed stable isotopes.

<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.

Tennessine is a synthetic chemical element; it has symbol Ts and atomic number 117. It has the second-highest atomic number and joint-highest atomic mass of all known elements, and is the penultimate element of the 7th period of the periodic table.

Flerovium is a superheavy synthetic chemical element; it has symbol Fl and atomic number 114. It is an extremely radioactive synthetic element, named after the Flerov Laboratory of Nuclear Reactions of the Joint Institute for Nuclear Research in Dubna, Russia, where the element was discovered in 1999. The lab's name, in turn, honours Russian physicist Georgy Flyorov. IUPAC adopted the name on 30 May 2012. The name and symbol had previously been proposed for element 102 (nobelium), but was not accepted by IUPAC at that time.

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

A nuclide is a class of atoms characterized by their number of protons, Z, their number of neutrons, N, and their nuclear energy state.

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).

Naturally occurring nickel (28Ni) is composed of five stable isotopes; 58
Ni
, 60
Ni
, 61
Ni
, 62
Ni
 and 64
Ni
, with 58
Ni
 being the most abundant. 26 radioisotopes have been characterised with the most stable being 59
Ni
 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.

Although there are nine known isotopes of helium (2He), only helium-3 and helium-4 are stable. All radioisotopes are short-lived, the longest-lived being 6
He
 with a half-life of 806.92(24) milliseconds. The least stable is 10
He
, with a half-life of 260(40) yoctoseconds, although it is possible that 2
He
 may have an even shorter half-life.

<span class="mw-page-title-main">Nuclear binding energy</span> Minimum energy required to separate particles within a nucleus

Nuclear binding energy in experimental physics is the minimum energy that is required to disassemble the nucleus of an atom into its constituent protons and neutrons, known collectively as nucleons. The binding energy for stable nuclei is always a positive number, as the nucleus must gain energy for the nucleons to move apart from each other. Nucleons are attracted to each other by the strong nuclear force. In theoretical nuclear physics, the nuclear binding energy is considered a negative number. In this context it represents the energy of the nucleus relative to the energy of the constituent nucleons when they are infinitely far apart. Both the experimental and theoretical views are equivalent, with slightly different emphasis on what the binding energy means.

<span class="mw-page-title-main">Valley of stability</span> Characterization of nuclide stability

In nuclear physics, the valley of stability is a characterization of the stability of nuclides to radioactivity based on their binding energy. Nuclides are composed of protons and neutrons. The shape of the valley refers to the profile of binding energy as a function of the numbers of neutrons and protons, with the lowest part of the valley corresponding to the region of most stable nuclei. The line of stable nuclides down the center of the valley of stability is known as the line of beta stability. The sides of the valley correspond to increasing instability to beta decay. The decay of a nuclide becomes more energetically favorable the further it is from the line of beta stability. The boundaries of the valley correspond to the nuclear drip lines, where nuclides become so unstable they emit single protons or single neutrons. Regions of instability within the valley at high atomic number also include radioactive decay by alpha radiation or spontaneous fission. The shape of the valley is roughly an elongated paraboloid corresponding to the nuclide binding energies as a function of neutron and atomic numbers.

Nickel-62 is an isotope of nickel having 28 protons and 34 neutrons.

<span class="mw-page-title-main">Atomic nucleus</span> Core of an atom; composed of nucleons (protons and neutrons)

The atomic nucleus is the small, dense region consisting of protons and neutrons at the center of an atom, discovered in 1911 by Ernest Rutherford based on the 1909 Geiger–Marsden gold foil experiment. After the discovery of the neutron in 1932, models for a nucleus composed of protons and neutrons were quickly developed by Dmitri Ivanenko and Werner Heisenberg. An atom is composed of a positively charged nucleus, with a cloud of negatively charged electrons surrounding it, bound together by electrostatic force. Almost all of the mass of an atom is located in the nucleus, with a very small contribution from the electron cloud. Protons and neutrons are bound together to form a nucleus by the nuclear force.

<span class="mw-page-title-main">Nuclear drip line</span> Atomic nuclei decay delimiter

The nuclear drip line is the boundary beyond which atomic nuclei are unbound with respect to the emission of a proton or neutron.

<span class="mw-page-title-main">Even and odd atomic nuclei</span> Nuclear physics classification method

In nuclear physics, properties of a nucleus depend on evenness or oddness of its atomic number Z, neutron number N and, consequently, of their sum, the mass number A. Most importantly, oddness of both Z and N tends to lower the nuclear binding energy, making odd nuclei generally less stable. This effect is not only experimentally observed, but is included in the semi-empirical mass formula and explained by some other nuclear models, such as the nuclear shell model. This difference of nuclear binding energy between neighbouring nuclei, especially of odd-A isobars, has important consequences for beta decay.

Unbiquadium, also known as element 124 or eka-uranium, is a hypothetical chemical element; it has placeholder symbol Ubq and atomic number 124. Unbiquadium and Ubq are the temporary IUPAC name and symbol, respectively, until the element is discovered, confirmed, and a permanent name is decided upon. In the periodic table, unbiquadium is expected to be a g-block superactinide and the sixth element in the 8th period. Unbiquadium has attracted attention, as it may lie within the island of stability, leading to longer half-lives, especially for 308Ubq which is predicted to have a magic number of neutrons (184).

Unbihexium, also known as element 126 or eka-plutonium, is a hypothetical chemical element; it has atomic number 126 and placeholder symbol Ubh. Unbihexium and Ubh are the temporary IUPAC name and symbol, respectively, until the element is discovered, confirmed, and a permanent name is decided upon. In the periodic table, unbihexium is expected to be a g-block superactinide and the eighth element in the 8th period. Unbihexium has attracted attention among nuclear physicists, especially in early predictions targeting properties of superheavy elements, for 126 may be a magic number of protons near the center of an island of stability, leading to longer half-lives, especially for 310Ubh or 354Ubh which may also have magic numbers of neutrons.

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.
This page is based on this Wikipedia article
Text is available under the CC BY-SA 4.0 license; additional terms may apply.
Images, videos and audio are available under their respective licenses.