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A180638
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Phan Thành Nam's upper bound on the number of non-relativistic electrons bound to a nucleus of charge n.
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1
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4, 6, 8, 9, 11, 12, 14, 15, 17, 18, 20, 21, 23, 24, 25, 27, 28, 30, 31, 32, 34, 35, 36, 38, 39, 40, 42, 43, 44, 46, 47, 48, 50, 51, 52, 54, 55, 56, 58, 59, 60, 62, 63, 64, 65, 67, 68, 69, 71, 72, 73, 75, 76, 77, 78, 80, 81, 82, 84, 85, 86, 87, 89, 90, 91, 93, 94, 95, 96, 98, 99, 100, 102
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OFFSET
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1,1
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COMMENTS
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From the abstract: "We prove that the maximal number N_c of non-relativistic electrons that a nucleus of charge Z can bind is less than 1.22Z + 3.1Z^(1/3).
This improves Lieb's (1984) upper bound N_c < 2Z+1 when Z <= 6. Our method also applies to non-relativistic atoms in magnetic field and to pseudo-relativistic atoms. We show that in these cases, under appropriate conditions, limsup_{Z->infty} N_c/Z <= 1.22."
Note that the published version of the paper, as revisions v2 and v3 on the arXiv, use the stronger bound 1.22Z + 3Z^(1/3) instead, where the constant 1.22 can be improved to 1.21684.... In particular this improves the bound for lithium, aluminium, scandium, chromium, cobalt, zinc, arsenic, krypton, yttrium, zirconium, etc. Asymptotically Seco, Sigal, & Solovej show that the maximum ionization is n + O(n^(5/7)) and it is conjectured that the maximum ionization is n + O(1). - Charles R Greathouse IV, Sep 13 2016
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REFERENCES
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E. H. Lieb and R. Seiringer, The stability of matter in quantum mechanics, Cambridge University Press, 2009.
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LINKS
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FORMULA
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a(n) = floor(1.22*n + 3.1*n^(1/3)).
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EXAMPLE
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a(5) = floor(1.22*5 + 3.1*5^(1/3)) = floor(11.40...) = 11, which means that boron (the element with atomic number 5, i.e., with 5 protons) can have no more than 11 bound electrons, which would give it a -6 charge. B^5- has been observed (in Al_3BC) so this bound is reasonably tight. - Charles R Greathouse IV, Sep 14 2016
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MAPLE
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MATHEMATICA
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PROG
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CROSSREFS
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KEYWORD
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easy,nonn
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AUTHOR
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EXTENSIONS
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Reference converted to link, entries checked - R. J. Mathar, Oct 06 2010
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STATUS
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approved
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