A Unified Explanation of Magic Numbers in Small Clusters

  • G. S. Anagnostatos


Small clusters with specific numbers of atoms, called magic numbers, exhibit particular stability properties in comparison to other cluster sizes. The identification of magic numbers in the literature, however, is rather confusing, since different sets of such numbers have been reported for different kind of atoms. For example, for Xe clusters1 the numbers 13, 19, 25, 55, 71, 87, 147, while for Ar and Kr the numbers2 14, 16, 19, 21, 23, 27, and 14, 16, 19, 22, 25, 27, 29, 39, 75, 87, respectively, and for Na the numbers3 2, 8, 20, 40, 58, 92 have been reported. The theoretical explanation of magic numbers in the literature is not unique as well. It follows two independent reasonings, one3 for the last set of above numbers (Na clusters; magic numbers as a result of decoupled 3s electrons driven by a central potential) and another1 for the other three sets (Xe, Ar, Kr clusters; magic numbers as a result of packing of spheres). Moreover, even for the same kind of clusters, theoretically expected magic numbers are not supported experimentally and experimentally observed numbers are not covered by theoryl.


Small Cluster Hard Sphere Magic Number Equilibrium Geometry Unify Explanation 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. 1.
    O. Echt, K. Sattler, and E. Recknagel, Magic numbers for sphere packings: Experimental verification in free xenon clusters, Phys. Rev. Lett. 47:1121(1981).ADSCrossRefGoogle Scholar
  2. 2.
    A. Ding and J. Hesslich, The abundance of Ar and Kr micro-clusters generated by supersonic expansion, Chem, Phys. Lett. 94:54(1983).ADSCrossRefGoogle Scholar
  3. 3.
    W. D. Knight, K. Clemenger, W.A. de Heer, W.A. Saunders, M. Y. Chow, and M. L. Cohen, Electronic shell structure and abundances of sodium clusters, Phys. Rev. Lett. 52:2141(1984).ADSCrossRefGoogle Scholar
  4. 4.
    J. A. Barker, The geometries of soft-sphere packings, J. Phys. (Paris), Collog. 38:C2–37(1977).Google Scholar
  5. 5.
    G. S. Anagnostatos, Isomorphic shell model for closed-shell nuclei, Int. J. Theor. Phys. 24:579(1985).CrossRefGoogle Scholar
  6. 6.
    J. Leech, Equilibrium of sets of particles on a sphere, Math. Gazelle 41:81(1957).MathSciNetCrossRefMATHGoogle Scholar
  7. 7.
    H. S. M. Coxeter, “Regular Polytopes” The MacMillan Co., New York (1973).Google Scholar
  8. 8.
    M. M. Kappes, R. W. Kunz, and E. Schumacker, Production of large sodium clusters (Nax, x ≤ 65) by seeded beam expansions, Chem. Phys. Lett. 91:413(1982).ADSCrossRefGoogle Scholar

Copyright information

© Plenum Press, New York 1987

Authors and Affiliations

  • G. S. Anagnostatos
    • 1
  1. 1.Tandem Accelerator LaboratoryNational Research Center of Natural Sciences “Demokritos”Aghia Paraskevi-AttikiGreece

Personalised recommendations