Speaker
Description
A question of the optimal configuration of a finite number of particles in a plane has been a difficult problem of both physics and mathematics for many centuries. Back in 1611, Kepler posed already the question of why a snowflake has perfect hexagonal symmetry [1]. At present, increased interest in the problem of the optimal configuration in a plane is also due to the development of nanotechnologies which make it possible to form systems of similarly charged particles confined by external potentials with a high symmetry. In particular, one of the important achievements of modern technology consists in the creation of «artificial atoms» or quantum dots, where a finite number of electrons is confined electrostatically in a nanometer-sized region [2].
In this communication we discuss the basic principles of self-organization of one-component charged particles, confined in disk and circular parabolic potentials. A system of equations is derived, that allows to determine equilibrium configurations for arbitrary, but finite, number of charged particles that are distributed over several rings [3,4]. The results of our approach demonstrate a remarkable agreement with the values provided by molecular dynamics calculations. With the increase of particle number n>180, we find a steady formation of a centered hexagonal lattice. At the same time, the energetic preferences for nonuniform local density then favor ground states where this locally hexagonal structure is isotropic dilated and contracted throughout the structure. In fact, the equilibrium configuration is determined by the need to achieve equilibrium through the formation of a hexagonal lattice on one side and a ring-like structure on the other. This competition leads to the formation of internal defects in such systems, in contrast to the case of unlimited regions, where the ground state of the system has no defects. Finally, this structure smoothly transforms to valence circular rings in the ground state configurations for both potentials. We briefly discuss the precursor of the phase transition of the type "hexagonal lattice - hexatic phase" with the increase of a particle number in the system at zero temperature [5].
References
1.J. Kepler, “The Six-Cornered Snowflake” (Clarendon, Oxford, 1966).
2.J. L. Birman, R. G. Nazmitdinov, V. I. Yukalov, Phys. Rep. 526, 1 (2013).
3.M. Cerkaski, R. G. Nazmitdinov, A. Puente, Phys. Rev. E 91, 032312 (2015).
4.R. G. Nazmitdinov, A. Puente, M. Cerkaski, M. Pons, Phys. Rev. E 95, 042603 (2017).
5.E. G. Nikonov, R. G. Nazmitdinov, P. I. Glukhovtsev, J. Surf. Investigation: X-ray, Synchrotron and Neutron Techniques 18, 248 (2024).
