Madelung Constants

The quantities obtained from cubic, hexagonal, etc., Lattice Sums, evaluated at $s=1$, are called Madelung constants. For cubic Lattice Sums, they are expressible in closed form for Even indices,

$$b_2(2) = -4\beta(1)\eta(1) =-4 {\textstyle{\pi\over 4}}\ln 2 = -\pi \ln 2$$ (1)

$$b_4(2) = -8\eta(1)\eta(0)=-8\ln 2\cdot{\textstyle{1\over 2}}=-4\ln 2.$$ (2)

$b_3(1)$ is given by Benson's Formula,

$$-b_3(1)=\setbox0=\hbox{$\scriptstyle{i,\, j,\, k=-\infty}$}\... ... sech}\nolimits ^2({\textstyle{1\over 2}}\pi\sqrt{m^2+n^2}\,),$$ (3)

where the prime indicates that summation over (0, 0, 0) is excluded. $b_3(1)$ is sometimes called ``the'' Madelung constant, corresponds to the Madelung constant for a 3-D NaCl crystal, and is numerically equal to $-1.74756\ldots$.

For hexagonal Lattice Sum, $h_2(2)$ is expressible in closed form as

$$h_2(2)=\pi\ln 3\sqrt{3}.$$ (4)

See also Benson's Formula, Lattice Sum

References

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. and Buhler, J. P. ``Elementary Function Expansions for Madelung Constants.'' J. Phys. Ser. A: Math. and Gen. 20, 5497-5510, 1987.

Finch, S. ``Favorite Mathematical Constants.'' http://www.mathsoft.com/asolve/constant/mdlung/mdlung.html