On Born’s Conjecture about Optimal Distribution of Charges for an Infinite Ionic Crystal



We study the problem for the optimal charge distribution on the sites of a fixed Bravais lattice. In particular, we prove Born’s conjecture about the optimality of the rock salt alternate distribution of charges on a cubic lattice (and more generally on a d-dimensional orthorhombic lattice). Furthermore, we study this problem on the two-dimensional triangular lattice and we prove the optimality of a two-component honeycomb distribution of charges. The results hold for a class of completely monotone interaction potentials which includes Coulomb-type interactions for \(d\ge 3\). In a more general setting, we derive a connection between the optimal charge problem and a minimization problem for the translated lattice theta function.


Calculus of variations Lattice energy Theta functions Electrostatic energy Ewald summation 

Mathematics Subject Classification

Primary 49S99 Secondary 82B20 



LB is grateful for the support of MATCH during his stay in Heidelberg. Both authors would like to thank Florian Nolte for interesting discussions.


  1. Aftalion, A., Blanc, X., Nier, F.: Lowest Landau level functional and Bargmann spaces for Bose–Einstein condensates. J. Funct. Anal. 241, 661–702 (2006)MathSciNetCrossRefMATHGoogle Scholar
  2. Alastuey, A., Jancovici, B.: On the classical two-dimensional one-component Coulomb plasma. J. Phys. 42(1), 1–12 (1981)MathSciNetCrossRefGoogle Scholar
  3. Antlanger, M., Kahl, G., Mazars, M., Samaj, L., Trizac, E.: Rich polymorphic behavior of Wigner bilayers. Phys. Rev. Lett. 117(11), 118002 (2016)CrossRefGoogle Scholar
  4. Assoud, L., Messina, R., Löwen, H.: Stable crystalline lattices in two-dimensional binary mixtures of dipolar particles. Europhys. Lett. 80(4), 1–6 (2007)CrossRefGoogle Scholar
  5. Bachman, G., Narici, L., Beckenstein, E.: Fourier and Wavelet Analysis. Springer, Berlin (2002)MATHGoogle Scholar
  6. Baernstein II, A.: A minimum problem for heat kernels of flat tori. Contemp. Math. 201, 227–243 (1997)MathSciNetCrossRefMATHGoogle Scholar
  7. Bernstein, S.: Sur les fonctions absolument monotones. Acta Math. 52, 1–66 (1929)MathSciNetCrossRefMATHGoogle Scholar
  8. Bétermin, L.: Local optimality of cubic lattices for interaction energies. Anal. Math. Phys. (2017).  https://doi.org/10.1007/s13324-017-0205-5
  9. Bétermin, L., Knüpfer, H.: Optimal lattice configurations for interacting spatially extended particles. Lett. Math. Phys. (2018).  https://doi.org/10.1007/s11005-018-1077-9
  10. Bétermin, L.: Two-dimensional theta functions and crystallization among Bravais lattices. SIAM J. Math. Anal. 48(5), 3236–3269 (2016)MathSciNetCrossRefMATHGoogle Scholar
  11. Bétermin, L., Petrache, M.: Dimension reduction techniques for the minimization of theta functions on lattices. J. Math. Phys. 58, 071902 (2017)MathSciNetCrossRefMATHGoogle Scholar
  12. Bétermin, L., Sandier, E.: Renormalized energy and asymptotic expansion of optimal logarithmic energy on the sphere. Constr. Approx. S. I. Approx. Stat. Phys. Part I 47(1), 39–74 (2018)MathSciNetMATHGoogle Scholar
  13. Bétermin, L., Zhang, P.: Minimization of energy per particle among Bravais lattices in \({\mathbb{R}}^2\): Lennard–Jones and Thomas–Fermi cases. Commun. Contemp. Math. 17(6), 1450049 (2015)MathSciNetCrossRefMATHGoogle Scholar
  14. Blanc, X., Lewin, M.: The crystallization conjecture: a review. EMS Surv. Math. Sci. 2, 255–306 (2015)MathSciNetCrossRefMATHGoogle Scholar
  15. Bochner, S.: Theta relations with spherical harmonics. Proc. Natl. Acad. Sci. USA 37(12), 804–808 (1951)MathSciNetCrossRefMATHGoogle Scholar
  16. Born, M.: Über elektrostatische Gitterpotentiale. Z. Phys. 7, 124–140 (1921)CrossRefGoogle Scholar
  17. Borwein, J., Glasser, M., McPhedran, R., Wan, J., Zucker, I.: Lattice Sums Then and Now (Encyclopedia of Mathematics and its Applications). Cambridge University Press, Cambridge (2013).  https://doi.org/10.1017/CBO9781139626804
  18. Bouman, N., Draisma, J., Van Leeuwaarden, J.S.H.: Energy minimization of repelling particles on a toric grid. SIAM J. Discrete Math. 27(3), 1295–1312 (2013)MathSciNetCrossRefMATHGoogle Scholar
  19. Cassels, J.W.S.: On a problem of Rankin about the Epstein zeta-function. Proc. Glasgow Math. Assoc. 4, 73–80 (1959)MathSciNetCrossRefMATHGoogle Scholar
  20. Cohn, H., Kumar, A.: Universally optimal distribution of points on spheres. J. Am. Math. Soc. 20(1), 99–148 (2007)MathSciNetCrossRefMATHGoogle Scholar
  21. Conway, J.H., Sloane, N.J.A.: Sphere Packings, Lattices and Groups, vol. 290. Springer, Berlin (1999)CrossRefMATHGoogle Scholar
  22. Coulangeon, R.: Spherical designs and zeta functions of lattices. Int. Math. Res. Not. 2006(16), 49620 (2006)MathSciNetMATHGoogle Scholar
  23. Coulangeon, R., Schürmann, A.: Local energy optimality of periodic sets. (Preprint) arXiv:1802.02072 (2018)
  24. Coulangeon, R., Lazzarini, G.: Spherical designs and heights of euclidean lattices. J. Number Theory 141, 288–315 (2014)MathSciNetCrossRefMATHGoogle Scholar
  25. Coulangeon, R., Schürmann, A.: Energy minimization, periodic sets and spherical designs. Int. Math. Res. Not. 4, 829–848 (2012)MathSciNetCrossRefMATHGoogle Scholar
  26. de Leeuw, S .W., Perram, J .W., Smith, E.R.: Simulation of electrostatic systems in periodic boundary conditions. I. Lattice sums and dielectric constants. Proc. R. Soc. Lon. A Math. Phys. Eng. Sci. 373(1752), 27–56 (1980)MathSciNetCrossRefGoogle Scholar
  27. De Luca, L., Friesecke, G.: Crystallization in two dimensions and a discrete Gauss–Bonnet theorem. J. Nonlinear Sci. 28(1), 69–90 (2018)MathSciNetCrossRefMATHGoogle Scholar
  28. Diananda, P.H.: Notes on two lemmas concerning the Epstein zeta-function. Proc. Glasgow Math. Assoc. 6, 202–204 (1964)MathSciNetCrossRefMATHGoogle Scholar
  29. E, W., Li, D.: On the crystallization of 2D hexagonal lattices. Commun. Math. Phys. 286, 1099–1140 (2009)Google Scholar
  30. Emersleben, O.: Zetafunktionen und elektrostatische Gitterpotentiale. I. Phys. Z. 24, 73–80 (1923)MATHGoogle Scholar
  31. Ennola, V.: A lemma about the Epstein zeta-function. Proc. Glasgow Math. Assoc. 6, 198–201 (1964)MathSciNetCrossRefMATHGoogle Scholar
  32. Epstein, P.: Zur Theorie allgemeiner Zetafunctionen. Math. Ann. 56(4), 615–644 (1903)MathSciNetCrossRefMATHGoogle Scholar
  33. Ewald, P.: Die Berechnung optischer und elektrostatischer Gitterpotentiale. Ann. Phys. 64, 253–287 (1921)CrossRefMATHGoogle Scholar
  34. Faulhuber, M., Steinerberger, S.: Optimal gabor frame bounds for separable lattices and estimates for Jacobi theta functions. J. Math. Anal. Appl. 445(1), 407–422 (2017)MathSciNetCrossRefMATHGoogle Scholar
  35. Flatley, L., Theil, F.: Face-centred cubic crystallization of atomistic configurations. Arch. Ration. Mech. Anal. 219(1), 363–416 (2015)CrossRefMATHGoogle Scholar
  36. Hardin, D.P., Saff, E.B., Simanek, Brian: Periodic discrete energy for long-range potentials. J. Math. Phys. 55(12), 123509 (2014)MathSciNetCrossRefMATHGoogle Scholar
  37. Hay, M.B., Workman, R.K., Manne, S.: Two-dimensional condensed phases from particles with tunable interactions. Phys. Rev. E 67(1), 012401 (2003)CrossRefGoogle Scholar
  38. Heitmann, R.C., Radin, C.: The ground state for sticky disks. J. Stat. Phys. 22, 281–287 (1980)MathSciNetCrossRefGoogle Scholar
  39. Henn, A.: The hexagonal lattice and the Epstein zeta function. In: Dynamical Systems, Number Theory and Applications, pp. 127–140 (2016).  https://doi.org/10.1142/9789814699877_0007
  40. Krazer, A., Prym, E.: Neue Grundlagen einer Theorie der Allgemeinen Theta-funktionen. Teubner, Leipzig (1893)MATHGoogle Scholar
  41. Levashov, V.A., Thorpe, M.F., Southern, B.W.: Charged lattice gas with a neutralizing background. Phys. Rev. B 67(22), 224109 (2003)CrossRefGoogle Scholar
  42. Mainini, E., Stefanelli, U.: Crystallization in carbon nanostructures. Commun. Math. Phys. 328, 545–571 (2014)MathSciNetCrossRefMATHGoogle Scholar
  43. Mainini, E., Piovano, P., Stefanelli, U.: Finite crystallization in the square lattice. Nonlinearity 27, 717–737 (2014)MathSciNetCrossRefMATHGoogle Scholar
  44. Montgomery, H.L.: Minimal theta functions. Glasgow Math. J. 30(1), 75–85 (1988)MathSciNetCrossRefMATHGoogle Scholar
  45. Mueller, E.J., Ho, T.-L.: Two-component Bose–Einstein condensates with a large number of vortices. Phys. Rev. Lett. 88(18), 180403 (2002)CrossRefGoogle Scholar
  46. Nonnenmacher, S., Voros, A.: Chaotic eigenfunctions in phase space. J. Stat. Phys. 92, 431–518 (1998)MathSciNetCrossRefMATHGoogle Scholar
  47. Osychenko, O.N., Astrakharchik, G.E., Boronat, J.: Ewald method for polytropic potentials in arbitrary dimensionality. Mol. Phys. 110(4), 227–247 (2012)CrossRefGoogle Scholar
  48. Pauling, L.: The principles determining the structure of complex ionic crystals. J. Am. Chem. Soc. 51(4), 1010–1026 (1929)CrossRefGoogle Scholar
  49. Perram, J.W., de Leeuw, S.W.: Statistical mechanics of two-dimensional coulomb systems. I. Lattice sums and simulation methodology. Phys. A 109(1–2), 237–250 (1981)MathSciNetCrossRefGoogle Scholar
  50. Radin, C.: The ground state for soft disks. J. Stat. Phys. 26(2), 365–373 (1981)MathSciNetCrossRefGoogle Scholar
  51. Rankin, R.A.: A minimum problem for the Epstein zeta-function. Proc. Glasgow Math. Assoc. 1, 149–158 (1953)MathSciNetCrossRefMATHGoogle Scholar
  52. Rougerie, N., Serfaty, S.: Higher dimensional coulomb gases and renormalized energy functionals. Commun. Pure Appl. Math. 69(3), 519–605 (2016)MathSciNetCrossRefMATHGoogle Scholar
  53. Samaj, L., Trizac, E.: Critical phenomena and phase sequence in a classical bilayer Wigner crystal at zero temperature. Phys. Rev. B 85(20), 205131 (2012)CrossRefGoogle Scholar
  54. Sandier, E., Serfaty, S.: From the Ginzburg–Landau model to vortex lattice problems. Commun. Math. Phys. 313(3), 635–743 (2012)MathSciNetCrossRefMATHGoogle Scholar
  55. Sarnak, P., Strömbergsson, A.: Minima of Epstein’s zeta function and heights of flat tori. Invent. Math. 165, 115–151 (2006)MathSciNetCrossRefMATHGoogle Scholar
  56. Schiff, J.L.: The Laplace transform: theory and applications. Springer, Berlin (2013)Google Scholar
  57. Stein, E.M., Shakarchi, R.: Complex Analysis. Princeton University Press, Princeton (2003)MATHGoogle Scholar
  58. Theil, F.: A proof of crystallization in two dimensions. Commun. Math. Phys. 262(1), 209–236 (2006)MathSciNetCrossRefMATHGoogle Scholar
  59. Ventevogel, W.J.: On the configuration of systems of interacting particle with minimum potential energy per particle. Phys. A Stat. Mech. Appl. 92A, 343 (1978)CrossRefGoogle Scholar
  60. Ventevogel, W.J., Nijboer, B.R.A.: On the configuration of systems of interacting particle with minimum potential energy per particle. Phys. A Stat. Mech. Appl. 98A, 274–288 (1979)MathSciNetCrossRefGoogle Scholar
  61. Xiao, Y., Thorpe, M.F., Parkinson, J.B.: Two-dimensional discrete coulomb alloy. Phys. Rev. B 59(1), 277–285 (1999)CrossRefGoogle Scholar

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© Springer Science+Business Media, LLC, part of Springer Nature 2018

Authors and Affiliations

  1. 1.QMATH, Department of Mathematical SciencesUniversity of CopenhagenCopenhagen ØDenmark
  2. 2.Institute of Applied Mathematics and IWRUniversity of HeidelbergHeidelbergGermany

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