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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,

$\displaystyle b_2(2)$ $\textstyle =$ $\displaystyle -4\beta(1)\eta(1) =-4 {\textstyle{\pi\over 4}}\ln 2 = -\pi \ln 2$ (1)
$\displaystyle b_4(2)$ $\textstyle =$ $\displaystyle -8\eta(1)\eta(0)=-8\ln 2\cdot{\textstyle{1\over 2}}=-4\ln 2.$ (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}\,),
\end{displaymath} (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}.
\end{displaymath} (4)

See also Benson's Formula, Lattice Sum


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.''

© 1996-9 Eric W. Weisstein