A181152 Decimal expansion of Madelung constant (negated) for the CsCl structure.

1, 7, 6, 2, 6, 7, 4, 7, 7, 3, 0, 7, 0, 9, 8, 8, 3, 9, 7, 9, 3, 5, 6, 7, 3, 3, 2, 0, 6, 3, 8, 6, 4, 4, 2, 9, 1, 1, 7, 0, 5, 2, 8, 6, 1, 9, 5, 8, 8, 5, 8, 5, 2, 8, 0, 6, 4, 9, 4, 1, 8, 4, 3, 7, 7, 2, 7, 9, 6, 6, 2, 2, 3, 7, 6, 9, 3, 4, 0, 8, 3, 0, 4, 7, 1, 5, 0, 9, 4, 5, 8, 1, 1, 2, 1, 6, 9, 8, 8, 9, 0, 8, 5, 6, 9

Offset: 1

Author

Leslie Glasser, Jan 24 2011

Keywords

• nonn
• cons

Comments

This is often quoted for a different lattice constant and multiplied by 2/sqrt(3) = 1.1547... = 10*A020832, which gives 1.76267...*1.1547... = 2.03536151... given in Zucker's Table 5 as the alpha for the CsCl structure, and by Sakamoto as the M_d for the B2 lattice. Given Zucker's b(1) = 0.774386141424002815... = A185577, this constant here is sqrt(3)*(3*b(1)+A085469)/4. - R. J. Mathar, Jan 28 2011

The CsCl structure consists of two interpenetrating simple cubic lattices of ions with charges +1 and -1, together occupying all the sites of the body-centered cubic lattice. - Andrey Zabolotskiy, Oct 21 2019

Links

• Leslie Glasser, Solid-State Energetics and Electrostatics: Madelung Constants and Madelung Energies, Inorg. Chem., 2012, 51 (4), 2420-2424.
• Y. Sakamoto, Madelung Constants of Simple Crystals Expressed in Terms of Born's Basic Potentials of 15 Figures, Journal of Chemical Physics, 28 (1958), 164-165. Errata: J. Chem. Phys, 28 (1958), 733; J. Chem. Phys, 28 (1958), 1253.
• Nicolas Tavernier, Gian Luigi Bendazzoli, Véronique Brumas, Stefano Evangelisti, and J. A. Berger, Clifford boundary conditions: a simple direct-sum evaluation of Madelung constants, J. Phys. Chem. Lett., 11 (2020), 7090-7095; arXiv:2006.01259 [physics.comp-ph], 2020.
• I. J. Zucker, Madelung constants and lattice sums for invariant cubic lattice complexes and certain tetragonal structures, J. Phys. A: Math. Gen. 8 (11) (1975) 1734.
• Wikipedia, Madelung constant

Crossrefs

Cf. A085469, A088537, A090734.

Programs

Mathematica

digits = 105;
m0 = 50; (* initial number of terms *)
dm = 10; (* number of terms increment *)
dd = 10; (* precision excess *)
Clear[f];
f[n_, p_] := f[n, p] = (s = Sqrt[n^2 + p^2]; ((2 + (-1)^n) Csch[s*Pi])/s // N[#, digits + dd]&);
f[m_] := f[m] = Pi/2 - (7 Log[2])/2 + 4 Sum[f[n, p], {n, 1, m}, {p, 1, m}];
f[m = m0];
f[m += dm];
While[Abs[f[m] - f[m - dm]] > 10^(-digits - dd), Print["f(", m, ") = ", f[m]]; m += dm];
A185577 = f[m];
Clear[g];
g[m_] := g[m] = 12 Pi Sum[Sech[(Pi/2) Sqrt[(2 j + 1)^2 + (2 k + 1)^2]]^2, {j, 0, m}, {k, 0, m}] // N[#, digits + dd]&;
g[m = m0];
g[m += dm];
While[Abs[g[m] - g[m - dm]] > 10^(-digits - dd), Print["g(", m, ") = ", g[m]]; m += dm];
A085469 = g[m];
A181152 = Sqrt[3] (A085469 - 3 A185577)/4;
RealDigits[A181152, 10, digits][[1]] (* Jean-François Alcover, May 07 2021 *)

Extensions

More terms (using the above comment from R. J. Mathar and terms from the b-files for A085469 and A185577) from Jon E. Schoenfield, Mar 10 2018

Definition corrected by Andrey Zabolotskiy, Oct 21 2019

a(88)-a(105) from Jean-François Alcover, May 07 2021