login
The OEIS is supported by the many generous donors to the OEIS Foundation.

 

Logo
Hints
(Greetings from The On-Line Encyclopedia of Integer Sequences!)
A085469 Decimal expansion of Madelung constant (negated) for NaCl structure. 20
1, 7, 4, 7, 5, 6, 4, 5, 9, 4, 6, 3, 3, 1, 8, 2, 1, 9, 0, 6, 3, 6, 2, 1, 2, 0, 3, 5, 5, 4, 4, 3, 9, 7, 4, 0, 3, 4, 8, 5, 1, 6, 1, 4, 3, 6, 6, 2, 4, 7, 4, 1, 7, 5, 8, 1, 5, 2, 8, 2, 5, 3, 5, 0, 7, 6, 5, 0, 4, 0, 6, 2, 3, 5, 3, 2, 7, 6, 1, 1, 7, 9, 8, 9, 0, 7, 5, 8, 3, 6, 2, 6, 9, 4, 6, 0, 7, 8, 8, 9, 9, 3 (list; constant; graph; refs; listen; history; text; internal format)
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.

Index entries for sequences related to f.c.c. lattice.

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

Cf. A004015, A005875, A108778 (continued fraction).

Sequence in context: A319883 A153186 A343628 * A050996 A085541 A133055

Adjacent sequences:  A085466 A085467 A085468 * A085470 A085471 A085472

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

Lookup | Welcome | Wiki | Register | Music | Plot 2 | Demos | Index | Browse | More | WebCam
Contribute new seq. or comment | Format | Style Sheet | Transforms | Superseeker | Recents
The OEIS Community | Maintained by The OEIS Foundation Inc.

License Agreements, Terms of Use, Privacy Policy. .

Last modified September 25 22:56 EDT 2022. Contains 356986 sequences. (Running on oeis4.)