A181152 - OEIS

%I #40 Oct 06 2021 03:28:48

%S 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,

%T 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,

%U 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

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

%C 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

%C 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

%H Leslie Glasser, <a href="https://doi.org/10.1021/ic2023852">Solid-State Energetics and Electrostatics: Madelung Constants and Madelung Energies</a>, Inorg. Chem., 2012, 51 (4), 2420-2424.

%H Y. Sakamoto, <a href="https://doi.org/10.1063/1.1744060">Madelung Constants of Simple Crystals Expressed in Terms of Born's Basic Potentials of 15 Figures</a>, Journal of Chemical Physics, 28 (1958), 164-165.  Errata: <a href="https://doi.org/10.1063/1.1744237">J. Chem. Phys, 28 (1958), 733</a>; <a href="https://doi.org/10.1063/1.1744387">J. Chem. Phys, 28 (1958), 1253</a>.

%H Nicolas Tavernier, Gian Luigi Bendazzoli, Véronique Brumas, Stefano Evangelisti, and J. A. Berger, <a href="https://doi.org/10.1021/acs.jpclett.0c01684">Clifford boundary conditions: a simple direct-sum evaluation of Madelung constants</a>, J. Phys. Chem. Lett., 11 (2020), 7090-7095; arXiv:<a href="https://arxiv.org/abs/2006.01259">2006.01259</a> [physics.comp-ph], 2020.

%H I. J. Zucker, <a href="https://doi.org/10.1088/0305-4470/8/11/008">Madelung constants and lattice sums for invariant cubic lattice complexes and certain tetragonal structures</a>, J. Phys. A: Math. Gen. 8 (11) (1975) 1734.

%H Wikipedia, <a href="https://en.wikipedia.org/wiki/Madelung_constant">Madelung constant</a>

%t digits = 105;

%t m0 = 50; (* initial number of terms *)

%t dm = 10; (* number of terms increment *)

%t dd = 10; (* precision excess *)

%t Clear[f];

%t f[n_, p_] := f[n, p] = (s = Sqrt[n^2 + p^2]; ((2 + (-1)^n) Csch[s*Pi])/s // N[#, digits + dd]&);

%t f[m_] := f[m] = Pi/2 - (7 Log[2])/2 + 4 Sum[f[n, p], {n, 1, m}, {p, 1, m}];

%t f[m = m0];

%t f[m += dm];

%t While[Abs[f[m] - f[m - dm]] > 10^(-digits - dd), Print["f(", m, ") = ", f[m]]; m += dm];

%t A185577 = f[m];

%t Clear[g];

%t 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]&;

%t g[m = m0];

%t g[m += dm];

%t While[Abs[g[m] - g[m - dm]] > 10^(-digits - dd), Print["g(", m, ") = ", g[m]]; m += dm];

%t A085469 = g[m];

%t A181152 = Sqrt[3] (A085469 - 3 A185577)/4;

%t RealDigits[A181152, 10, digits][[1]] (* _Jean-François Alcover_, May 07 2021 *)

%Y Cf. A085469, A088537, A090734.

%K nonn,cons

%O 1,2

%A _Leslie Glasser_, Jan 24 2011

%E 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

%E Definition corrected by _Andrey Zabolotskiy_, Oct 21 2019

%E a(88)-a(105) from _Jean-François Alcover_, May 07 2021