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A018226 Magic numbers of nucleons: nuclei with one of these numbers of either protons or neutrons are more stable against nuclear decay. 36
2, 8, 20, 28, 50, 82, 126 (list; graph; refs; listen; history; text; internal format)



In the shell model for the nucleus, magic numbers are the numbers of either protons or neutrons at which a shell is filled.

First seven positive terms of A162626. - Omar E. Pol, Jul 07 2009

Steppenbeck: "The results of the experiment indicate that 54Ca's first excited state lies at a relatively high energy, which is characteristic of a large nuclear shell gap, thus indicating that N = 34 in 54Ca is a new magic number, as predicted theoretically by the University of Tokyo group in 2001. By conducting a more detailed comparison to nuclear theory the researchers were able to show that the N = 34 magic number is equally as significant as some other nuclear shell gaps."


A brief description is given under "Magic numbers" in the Encyclopedia Britannica.

Dictionary of Science (Simon and Schuster), see the entry for "Magic number".


Table of n, a(n) for n=1..7.

Radoslav Jovanovic, Magic Numbers and the Pascal Triangle

S. Bjornholm, Clusters, condensed matter in embryonic form, Contemp. Phys. 31 1990 pp. 309-324 (p. 312).

J. Fridmann et al., 'Magic' nucleus 42-Si, Nature, 435 (2005), 922-924 and 897-898.

J. Glanz, Uut and Uup Add Their Atomic Mass to Periodic Table, New York Times, Feb 01, 2003, pages 1 and 26.

R. V. F. Janssens, Unexpected doubly magic nucleus, Nature, 459 (Jun 25 2009), 1069-1070. [_Added by N. J. A. Sloane, Jul 05 2009]

Lutvo Kuric, Digital nuclear shell model, International Letters of Chemistry, Physics and Astronomy, 13(2) (2014) 160-173; ISSN 2299-3843.

V. Ladma, Magic Numbers

NAPC Isotope Hydrology Section, Chapter 2, Atomic Systematics and Nuclear Structure [Broken link?]

R. Nave, Shell Model of Nucleus

R. Nave, Enhanced Abundance of Magic Number Nuclei

Rachele Nerattini, Johann S. Brauchart, Michael K.-H. Kiessling, Magic numbers in Smale's 7th problem, arXiv:1307.2834v1 [math-ph], July 10, 2013.

Phys.org, Evidence for a new nuclear 'magic number', Oct 9, 2013.

D. Steppenbeck et al., Evidence for a new nuclear 'magic number' from the level structure of 54Ca, Nature, 2013 DOI: 10.1038/nature12522.

D. Warner, Not-so-magic numbers</>, Nature, 430 (Jul 29 2004), 517-519.

D. Weise, The Pythagorean Approach to Problems of Periodicity in Chemistry and Nuclear Physics

Wikipedia, Magic number (physics)


If 1 <= n <= 3 then a(n)=n*(n+1)*(n+2)/3, else if 4 <= n <= 7 then a(n)=n(n^2+5)/3. - Omar E. Pol, Jul 07 2009 [This needs to be clarified. - Joerg Arndt, May 03 2011]

From Daniel Forgues, May 03 2011: (Start)

If 1 <= n <= 3 then a(n) = 2 T_n, else

if 4 <= n <= 7 then a(n) = 2 (T_n - t_{n-1}),

where T_n is the n-th tetrahedral number, t_n the n-th triangular number.

G.f.: (2*x*(1 - 6*x^3 + 14*x^4 - 11*x^5 + 3*x^6))/(1 - x)^4, 1 <= n <= 7.

Using those formulas for n >= 0 gives A162626. (End)

a(n) = n*(n^2+5)/3 + (4*n-6)*A171386(n). - Omar E. Pol, Aug 14 2013


Cf. A018227 Number of electrons (which equals number of protons) such that they are arranged into complete shells within the atom.

Cf. A033547, A110856, A130598, A162626, A046939, A046940.

Sequence in context: A192153 A190640 A230434 * A162626 A137306 A110856

Adjacent sequences:  A018223 A018224 A018225 * A018227 A018228 A018229




John Raithel (raithel(AT)rahul.net)



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Last modified October 23 05:56 EDT 2019. Contains 328335 sequences. (Running on oeis4.)