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ORAL CONTRIBUTIONS
Madelung Energy of Regular Lattices
P. C. N. Pereira and S. W. S. Apolinario
Departamento de Fisica, Universidade Federal de Pernambuco, 50670-901 Recife, PE, Brazil
Due to strong Coulomb repulsion, an electron gas at low temperature and
density undergoes a phase transition to a crystalline state called the Wigner crystal, as
predicted theoretically by Eugene Wigner in 1934. The Wigner crystal is one of the
possible ground states of a strongly correlated electron system. Wigner crystallization
was first observed in an electron gas confined above the surface of liquid helium by
Grimes and Adams. Several theoretical and experimental works have considered the
properties of a finite number of particles interacting through a repulsive interparticle
interaction potential and kept together by an external confinement potential. Typical
experimental realizations of such 2D systems include electrons on the surface of liquid
helium, vortices in a type II superconducting 2D films, colloidal suspensions, and dusty
plasma crystals. These systems and their configurations have been observed
experimentally, and depending of their conditions, they can form regular lattices.
We propose a method to compute the Madelung energy (ME) of regular lattices.
As a first test, such a method was applied for Yukawa lattices [1] of different topologies
and dimensions which resulted in an approximate closed form expression of the ME.
The obtained Madelung energies have a satisfactory accuracy for all ranges of the
screening parameter κ of the Yukawa potential, and it becomes exact in the asymptotic
limits κ→0 and κ→+∞. For instance, for the triangular lattice, the maximum relative
error of the ME given by the method is about 0.0047. Also, satisfactory results are
obtained for the one-component plasma limit. The Madelung constants of the twodimensional
hexagonal BN and square NaCl and the three-dimensional cubic NaCl
crystals are estimated with a relative error of 0.004, 0.006, and 0.03, respectively.
Finally, different ways to improve the method are presented and discussed.
[1] Medelung energy of Yukawa lattices, P. C. N. Pereira and S. W. S. Apolinario,
Phys. Rev. E 86, 046702 (2012).
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