A085469 -id:A085469 - cp's OEIS frontend

A247042 Decimal expansion of delta_2 (negated), a constant associated with a certain two-dimensional lattice sum.

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

Offset: 1

Jean-François Alcover, Sep 10 2014

This constant is named sigma(1/2) in the Borwein reference.

			-3.900264920001955882845475336604973219209047856477537388...

Steven R. Finch, Mathematical Constants, Cambridge University Press, 2003, Sections 1.10 Madelung's constant, p. 79.

D. Borwein, J. M. Borwein and R. Shail, Analysis of Certain Lattice Sums, Journal of Mathematical Analysis and Applications, Volume 143, Issue 1, October 1989, Pages 126-137.
Eric Weisstein's MathWorld, Lattice Sum
Eric Weisstein's MathWorld, Madelung Constants

Cf. A088537, A085469, A090734, A247040.

delta2 = 2*Zeta[1/2]*(Zeta[1/2, 1/4] - Zeta[1/2, 3/4]); RealDigits[delta2, 10, 102] // First

2*zeta(1/2)*(zetahurwitz(1/2,1/4)-zetahurwitz(1/2,3/4)) \\ Charles R Greathouse IV, Jan 31 2018

delta_2 = 2*zeta(1/2)*(zeta(1/2, 1/4) - zeta(1/2, 3/4)), where zeta(s,a) gives the generalized Riemann zeta function.

A182565 Decimal expansion of Madelung constant (negated) for cuprous oxide Cu_2O.

Original entry on oeis.org

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

Offset: 1

N. J. A. Sloane, May 05 2012

			-4.4424752098389...

Leslie Glasser, Solid-State Energetics and Electrostatics: Madelung Constants and Madelung Energies, Inorg. Chem., 2012, 51 (4), 2420-2424.
Yosio Sakamoto, Madelung Constants of Simple Crystals Expressed in Terms of Born's Basic Potentials of 15 Figure, The Journal of Chemical Physics 28, 164 (1958). See Table III, row C3, column M_r.
Wikipedia, Copper(I) oxide.

Cf. A085469, A181152, A182566, A182567, A185577, A185578.

Equals (sqrt(3)/4) * (2 * A085469 + 3 * (A185577 + A185578)). [Sakamoto] - Andrey Zabolotskiy, Oct 12 2023

More terms using the formula added by Andrey Zabolotskiy, Oct 12 2023

A182566 Decimal expansion of Madelung constant (negated) for zinc sulfide ZnS.

Original entry on oeis.org

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

Offset: 1

N. J. A. Sloane, May 05 2012

This constant is for sphalerite, the more common polymorph of zinc sulfide. - Jon E. Schoenfield, Mar 11 2018

The sphalerite (zincblende) structure consists of two interpenetrating face-centered cubic lattices of ions with charges +2 and -2, together occupying all sites of the diamond structure. This constant does not include overall factor 2 from the absolute value of the charges. This constant is normalized to the nearest neighbor distance, not the lattice constant; cf. the values 3.782926104 (or 3.78292610408577750878 from Sarkar & Bhattacharyya) and 7.56584 from pages 41 and 16 of "Lattice Sums Then and Now", which equal 4/sqrt(3) and 8/sqrt(3) times this constant. - Andrey Zabolotskiy, Feb 18 2021

			-1.6380550533887...

J. M. Borwein, M. L. Glasser, R. C. McPhedran, J. G. Wan, and I. J. Zucker, Lattice Sums Then and Now, Cambridge University Press, 2013.
Leslie Glasser, Solid-State Energetics and Electrostatics: Madelung Constants and Madelung Energies, Inorg. Chem., 2012, 51 (4), 2420-2424.
B. Sarkar and K. Bhattacharyya, Accurate calculation of Coulomb sums: Efficacy of Padé-like methods, Phys. Rev. B 48, 6913 (1993). See Table VI, row ZnS, column Regular.
Sebastian Steiger, Mehdi Salmani-Jelodar, Denis Areshkin, Abhijeet Paul, Tillmann Kubis, Michael Povolotskyi, Hong-Hyun Park and Gerhard Klimeck, Enhanced valence force field model for the lattice properties of gallium arsenide, Phys. Rev. B 84, 155204 (2011). See Eq. (18).
Nicolas Tavernier, Gian Luigi Bendazzoli, Véronique Brumas, Stefano Evangelisti, and J. A. Berger, Clifford boundary conditions: a simple direct-sum evaluation of Madelung constants, arXiv:2006.01259 [physics.comp-ph], 2020.
Wikipedia, Madelung constant.
Wikipedia, Zinc sulfide.

Cf. A085469, A181152, A182565, A182567.

Equals (sqrt(3)/4) * A085469 + (1/2) * A181152. [Glasser] - Andrey Zabolotskiy, Oct 12 2023

More terms based on Sarkar & Bhattacharyya and Steiger et al. added by Andrey Zabolotskiy, Jul 16 2023

More terms using the formula added by Andrey Zabolotskiy, Oct 12 2023

A182567 Decimal expansion of Madelung constant (negated) for calcium fluoride CaF_2.

Original entry on oeis.org

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

Offset: 1

N. J. A. Sloane, May 05 2012

			-5.0387848798...

Leslie Glasser, Solid-State Energetics and Electrostatics: Madelung Constants and Madelung Energies, Inorg. Chem., 2012, 51 (4), 2420-2424.
Wikipedia, Fluorite structure.

Cf. A085469, A181152, A182565, A182566.

This constant = sqrt(3)/2 * A085469 + 2 * A181152. [Glasser] - Andrey Zabolotskiy, Oct 12 2023

More terms using the formula added by Andrey Zabolotskiy, Oct 12 2023

A264156 Decimal expansion of M_5, the 5-dimensional analog of Madelung's constant (negated).

Original entry on oeis.org

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

Offset: 1

Jean-François Alcover, Nov 06 2015

			-1.9093378156187685595201437984336...

Steven R. Finch, Mathematical Constants, Cambridge University Press, 2003, Sections 1.10 Madelung's constant, p. 77.

Eric Weisstein's World of Mathematics, Madelung Constants.

Cf. A088537, A085469, A090734, A247040, A261805, A264157.

digits = 32; f[n_, x_] := 1/Sqrt[Pi*x]*(EllipticTheta[4, 0, Exp[-x]]^n - 1); M[5] = NIntegrate[f[5, x], {x, 0, Infinity}, WorkingPrecision -> digits + 5]; RealDigits[M[5], 10, digits] // First

Equals (1/sqrt(Pi))*Integral_{t=0..oo} ((Sum_{k=-oo..oo} (-1)^k*exp(-k^2*t))^5 - 1)/sqrt(t) dt.

A264157 Decimal expansion of M_7, the 7-dimensional analog of Madelung's constant (negated).

Original entry on oeis.org

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

Offset: 1

Jean-François Alcover, Nov 06 2015

			-2.01240598979798606439503063580430044165678065812192932878490469117330...

Steven R. Finch, Mathematical Constants, Cambridge University Press, 2003, Sections 1.10 Madelung's constant, p. 77.

Eric Weisstein's World of Mathematics, Madelung Constants.

Cf. A088537, A085469, A090734, A247040, A261805, A264156.

digits = 32; f[n_, x_] := 1/Sqrt[Pi*x]*(EllipticTheta[4, 0, Exp[-x]]^n - 1); M[7] = NIntegrate[f[7, x], {x, 0, Infinity}, WorkingPrecision -> digits + 5]; RealDigits[M[7], 10, digits] // First

th4(x)=1+2*sumalt(n=1,(-1)^n*x^n^2)
intnum(x=0,[oo,1], (th4(exp(-x))^7-1)/sqrt(Pi*x)) \\ Charles R Greathouse IV, Jun 06 2016

Equals (1/sqrt(Pi))*Integral_{t=0..oo} ((Sum_{k=-oo..oo} (-1)^k*exp(-k^2*t))^7-1)/sqrt(t) dt.

More terms from Charles R Greathouse IV, Jun 06 2016

A336275 Decimal expansion of the dimensionless coefficient of the Coulomb self-energy of a uniformly charged two-dimensional square.

Original entry on oeis.org

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

Offset: 1

Amiram Eldar, Jul 15 2020

Coulomb self-energy of a system of electric charges is the total electrostatic potential energy of interaction between charge elements.

For a uniformly charged two-dimensional square with a total charge Q and a side length L it is equal to c * k*Q^2/L, where k is the Coulomb constant (A182999) and c is this constant.

			1.486604799123689351264092833819785899009872739605305...

Orion Ciftja, Coulomb self-energy and electrostatic potential of a uniformly charged square in two dimensions, Physics Letters A, Vol. 374, No. 7 (2010), pp. 981-983, alternative link.
Wikipedia, Electric potential energy.

Cf. A085469, A182999, A336274.

RealDigits[2*(1 - Sqrt[2])/3 + 2*Log[1 + Sqrt[2]], 10, 100][[1]]

2*(1 - sqrt(2))/3 + 2 * log(1 + sqrt(2)) \\ Michel Marcus, Jul 15 2020

Equals 2*(1 - sqrt(2))/3 + 2 * log(1 + sqrt(2)).

A246966 Decimal expansion of H_2, the analog of Madelung's constant for the planar hexagonal lattice.

Original entry on oeis.org

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

Offset: 1

Jean-François Alcover, Sep 10 2014

The ionic hexagonal (triangular) lattice considered here consists of three interpenetrating hexagonal lattices of ions with charges +1, -1, 0. Equivalently, one may consider the honeycomb net consisting of two hexagonal lattices of ions with charges +1 and -1 (the h-BN layer structure). In any case, this lattice sum is based on the nearest neighbor distance (not the length of the period of the ionic crystal structure, which is sqrt(3) times greater). - Andrey Zabolotskiy, Jun 21 2022

			1.54221972170650525853141576436424529561948...

Steven R. Finch, Mathematical Constants, Cambridge University Press, 2003, Sections 1.10 Madelung's constant, p. 78.

David Borwein, Jonathan M. Borwein and Keith F. Taylor, Convergence of lattice sums and Madelung's constant, J. Math. Phys. 26 (1985), 2999-3009.
Eric Weisstein's MathWorld, Madelung Constants

Cf. A088537, A085469, A090734, A247040.

H2 = (-3 + Sqrt[3])*Zeta[1/2]*((1 - Sqrt[2])*Zeta[1/2, 1/3] + Zeta[1/2, 1/6]); RealDigits[H2, 10, 103] // First

(sqrt(3)-3)*zeta(1/2)*((1-sqrt(2))*zetahurwitz(1/2, 1/3) + zetahurwitz(1/2, 1/6)) \\ Charles R Greathouse IV, Jan 31 2018

H_2 = (-3 + sqrt(3))*zeta(1/2)*((1 - sqrt(2))*zeta(1/2, 1/3) + zeta(1/2, 1/6)), where zeta(s,a) gives the generalized Riemann zeta function.

A271872 Decimal expansion of the doubly infinite sum N_3 = Sum_{i,j,k = -inf..inf} (-1)^(i+j+k)/(i^2+j^2+k^2), a lattice constant analog of Madelung's constant (negated).

Original entry on oeis.org

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

Offset: 1

Jean-François Alcover, Apr 24 2016

			-2.51935615208944531334271172732943791211649913675173257750066...

Steven R. Finch, Mathematical Constants, Cambridge University Press, 2003, Section 1.10 Madelung's constant, p. 77.

Eric Weisstein's MathWorld, Madelung Constants

Cf. A088537 (M_2), A085469 (M_3), A090734 (M_4), A086054 (N_2).

digits = 101; Clear[s]; s[max_] := s[max] = NSum[(-1)^n Csch[Pi *Sqrt[m^2 + 2 n^2]]/Sqrt[m^2 + 2 n^2], {m, 1, max}, {n, 1, max}, Method -> "AlternatingSigns", WorkingPrecision -> digits + 10]; s[10]; s[max = 20]; Print[max]; While[RealDigits[s[max], 10, digits + 5][[1]] != RealDigits[s[max/2], 10, digits + 5][[1]], max = max*2; Print[max]]; N3 = Pi^2/3 - Pi*Log[2] - Pi/Sqrt[2] Log[2 (Sqrt[2] + 1)] + 8 Pi*s[max]; RealDigits[N3, 10, digits][[1]]

N_3 = Pi^2/3-Pi*log(2)-(Pi/sqrt(2))*log(2(sqrt(2)+1))+8 Pi*Sum_{m,n >= 1} (-1)^n csch(Pi*sqrt(m^2+2n^2))/sqrt(m^2+2n^2).

cp's OEIS Frontend

A247042 Decimal expansion of delta_2 (negated), a constant associated with a certain two-dimensional lattice sum.

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A182565 Decimal expansion of Madelung constant (negated) for cuprous oxide Cu_2O.

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A182566 Decimal expansion of Madelung constant (negated) for zinc sulfide ZnS.

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A182567 Decimal expansion of Madelung constant (negated) for calcium fluoride CaF_2.

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A264156 Decimal expansion of M_5, the 5-dimensional analog of Madelung's constant (negated).

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A264157 Decimal expansion of M_7, the 7-dimensional analog of Madelung's constant (negated).

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A336275 Decimal expansion of the dimensionless coefficient of the Coulomb self-energy of a uniformly charged two-dimensional square.

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A246966 Decimal expansion of H_2, the analog of Madelung's constant for the planar hexagonal lattice.

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