Madelung Constants

The quantities obtained from cubic, hexagonal, etc., lattice sums, evaluated at s=1, are called Madelung constants.

For cubic lattice sums


the Madelung constants expressible in closed form for even indices n, a few examples of which are summarized in the following table, where beta(n) is the Dirichlet beta function and eta(n) is the Dirichlet eta function.


To obtain the closed form for b_2(s), break up the double sum into pieces that do not include i=j=0,


where the negative sums have been reindexed to run over positive quantities. But (-1)^i=(-1)^(-i), so all the above terms can be combined into


The second of these sums can be done analytically as


which in the case s=1 reduces to


The first sum is more difficult, but in the case s=1 can be written


Combining these then gives the original sum as


b_3(1) is given by Benson's formula (Borwein and Bailey 2003, p. 24)


(OEIS A085469), where the prime indicates that summation over (0, 0, 0) is excluded.

b_3(1)=M is sometimes called "the" Madelung constant, corresponds to the Madelung constant for a three-dimensional NaCl crystal. Crandall (1999) gave the expression

 +2sum^'_(m, n, p=-infty)^infty((-1)^(m+n+p))/(sqrt(m^2+n^2+p^2)[exp(4pisqrt(m^2+n^2+p^2))+1]).

Similar results were found by Tyagi (2004),


the last of which converges rapidly. Averaging (16) and (13) then gives the beautiful equation


which is correct to 10 decimal digits even if the sum is completely omitted (Tyagi 2004).

However, no closed form for b_3(1) is known (Bailey et al. 2006).

For hexagonal lattice sums, h_2(2) is expressible in closed form as


(OEIS A086055).

See also

Benson's Formula, Harmonic Series, Lattice Sum

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Bailey, D. H.; Borwein, J. M.; Crandall, R. E.; and Zucker, I. J. "Lattice Sums Arising from the Poisson Equation." J. Phys. A 46, 115201, 31 pp., 2013.Bailey, D. H.; Borwein, J. M.; Kapoor, V.; and Weisstein, E. W. "Ten Problems in Experimental Mathematics." Amer. Math. Monthly 113, 481-509, 2006.Borwein, J. and Bailey, D. Mathematics by Experiment: Plausible Reasoning in the 21st Century. Wellesley, MA: A K Peters, 2003.Borwein, J. M. and Borwein, P. B. Pi & the AGM: A Study in Analytic Number Theory and Computational Complexity. New York: Wiley, 1987.Buhler, J. and Wagon, S. "Secrets of the Madelung Constant." Mathematica in Education and Research 5, 49-55, Spring 1996.Crandall, R. E. "New Representations for the Madelung Constant." Exp. Math. 8, 367-379, 1999.Crandall, R. E. and Buhler, J. P. "Elementary Function Expansions for Madelung Constants." J. Phys. Ser. A: Math. and Gen. 20, 5497-5510, 1987.Finch, S. R. Mathematical Constants. Cambridge, England: Cambridge University Press, 2003.Havil, J. "Madelung's Constant." §3.4 in Gamma: Exploring Euler's Constant. Princeton, NJ: Princeton University Press, pp. 33-35, 2003.Sloane, N. J. A. Sequences A016639, A085469, A086054, and A086055 in "The On-Line Encyclopedia of Integer Sequences."Tyagi, S. "New Series Representation for Madelung Constant." Oct. 17, 2004.

Referenced on Wolfram|Alpha

Madelung Constants

Cite this as:

Weisstein, Eric W. "Madelung Constants." From MathWorld--A Wolfram Web Resource.

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