OFFSET
1,2
COMMENTS
This is the electrostatic potential at the origin produced by unit charges of sign (-1)^(i+j+k) at all nonzero lattice points (i,j,k).
The NaCl structure consists of two interpenetrating face-centered cubic lattices of ions with charges +1 and -1, together occupying all the sites of the simple cubic lattice. - Andrey Zabolotskiy, Oct 21 2019
Named after the German physicist Erwin Madelung (1881-1972). - Amiram Eldar, Apr 02 2022
REFERENCES
Richard E. Crandall, Topics in Advanced Scientific Computation, Springer, Telos books, 1996, pages 73-79.
Steven R. Finch, Mathematical Constants, Cambridge, 2003, p. 76.
Sadri Hassani, Mathematical Methods Using Mathematica: For Students of Physics and Related Fields, Springer, NY, page 60.
LINKS
Jean-François Alcover, Table of n, a(n) for n = 1..2003 [There were errors in the previous b-file, which had 1847 terms contributed by Harry J. Smith.]
David H. Bailey, Jonathan M. Borwein, Vishaal Kapoor and Eric W. Weisstein, Ten Problems in Experimental Mathematics, Amer. Math. Monthly, Vol. 113, No. 6 (2006), pp. 481-509.
R. E. Crandall and J. P. Buhler, Elementary function expansions for Madelung constants, J. Phys. A: Math. Gen., Vol. 20, No. 16 (1987), pp. 5497-5510.
R. E. Crandall and J. P. Buhler, The potential within a crystal lattice, J. Phys. A: Math. Gen., Vol. 20, No. 9 (1987), pp. 2279-2292.
E. R. Fuller, Jr. and E. R. Naimon, Electrostatic Contributions to the Brugger-Type Elastic Constants, Phys. Rev. B, Vol. 6, No. 10 (1971), pp. 3609-3620.
Leslie Glasser, Solid-State Energetics and Electrostatics: Madelung Constants and Madelung Energies, Inorg. Chem., Vol. 51, No. 4 (2012), pp. 2420-2424; DOI: 10.1021/ic2023852.
André Hautot, New applications of Poisson's summation formula, J. of Phys. A, Vol. 8, No. 6 (1975) pp. 853-862.
Simon Plouffe, Madelung constant.
Simon Plouffe, The Levy constant.
Nicolas Tavernier, Gian Luigi Bendazzoli, Véronique Brumas, Stefano Evangelisti, and J. A. Berger, Clifford boundary conditions: a simple direct-sum evaluation of Madelung constants, arXiv:2006.01259 [physics.comp-ph], 2020.
Nicolas Tavernier, Gian Luigi Bendazzoli, Véronique Brumas, Stefano Evangelisti, and J. Arjan Berger, Clifford Boundary Conditions for Periodic Systems: the Madelung Constant of Cubic Crystals in 1, 2 and 3 Dimensions, arXiv:2107.04686 [cond-mat.mtrl-sci], 2021.
Sandeep Tyagi, New series representation of the Madelung constant, Prog. Theor. Phys., Vol. 114, No. 3 (2005), pp. 517-521.
Eric Weisstein's World of Mathematics, Benson's Formula.
Eric Weisstein's World of Mathematics, Madelung Constants.
Wikipedia, Madelung constant.
FORMULA
Sum_{i, j, k not all 0} (-1)^(i+j+k)/sqrt(i^2+j^2+k^2).
EXAMPLE
-1.7475645946331821906362120355443974034851614366247417581528253507...
MATHEMATICA
RealDigits[ 12Pi*Sum[ Sech[Pi/2*Sqrt[(2j + 1)^2 + (2k + 1)^2]]^2, {j, 0, 40}, {k, 0, 40}], 10, 111][[1]] (* Robert G. Wilson v, Jul 12 2005 *)
RealDigits[Quiet[12 Pi (Sech[Pi/Sqrt[2]]^2 + NSum[Sum[Sech[Pi Norm[2 v + 1]/2]^2, {v, FrobeniusSolve[{1, 1}, Round[m]]}, Method -> "Procedural"], {m, 1, Infinity}, Compiled -> False, Method -> "WynnEpsilon", NSumTerms -> 33, WorkingPrecision -> 100])]][[1]] (* Jan Mangaldan, Jun 25 2020 *)
digits = 1800; m0 = 800; dm = 10; dd = 10; Clear[f, g];
g[j_, k_] := g[j, k] = 12 Pi Sech[(Pi/2) Sqrt[(2j + 1)^2 + (2k + 1)^2]]^2 // N[#, digits + dd]&;
f[m_] := f[m] = Sum[g[j, k], {j, 0, m}, {k, 0, m}];
f[m = m0]; f[m += dm];
While[Abs[f[m] - f[m - dm]] > 10^(-digits - dd), Print[m]; m += dm];
A085469 = f[m];
RealDigits[A085469, 10, digits][[1]] (* Jean-François Alcover, May 08 2021, after Robert G. Wilson v *)
CROSSREFS
KEYWORD
nonn,cons
AUTHOR
Eric W. Weisstein, Jul 01 2003
EXTENSIONS
Entry revised by N. J. A. Sloane, Apr 12, 2004
Definition corrected by Leslie Glasser, Jan 24 2011
Definition corrected by Andrey Zabolotskiy, Oct 21 2019
STATUS
approved