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Symmetry 2018, 10(6), 228; https://doi.org/10.3390/sym10060228

Harmonic Principles of Elemental Crystals—From Atomic Interaction to Fundamental Symmetry

1
Institute of Experimental Physics, TU Bergakademie Freiberg, Leipziger Str. 23, 09599 Freiberg, Germany
2
Samara Center for Theoretical Materials Science, Samara National Research University, Ac. Pavlov Street 1, 443011 Samara, Russia
3
Cinector GmbH, Poststr. 25, 09648 Mittweida, Germany
*
Author to whom correspondence should be addressed.
Received: 5 June 2018 / Revised: 8 June 2018 / Accepted: 11 June 2018 / Published: 15 June 2018
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Abstract

The formation of crystals and symmetry on the atomic scale has persistently attracted scientists through the ages. The structure itself and its subtle dependence on boundary conditions is a reflection of three principles: atomic attraction, repulsion, and the limitations in 3D space. This involves a competition between simplicity and high symmetry on the one hand and necessary structural complexity on the other. This work presents a simple atomistic crystal growth model derived for equivalent atoms and a pair potential. It highlights fundamental concepts, most prominently provided by a maximum number of equilibrium distances in the atom’s local vicinity, to obtain high symmetric structural motifs, among them the Platonic Solids. In this respect, the harmonically balanced interaction during the atomistic nucleation process may be regarded as origin of symmetry. The minimization of total energy is generalized for 3D periodic structures constituting these motifs. In dependence on the pair potential’s short- and long-range characteristics the, by symmetry, rigid lattices relax isotropically within the potential well. The first few coordination shells with lattice-specific fixed distances do not necessarily determine which equilibrium symmetry prevails. A phase diagram calculated on the basis of these few assumptions summarizes stable regions of close-packed fcc and hcp, next to bcc symmetry for predominantly soft short-range and hard long-range interaction. This lattice symmetry, which is evident for alkali metals as well as transition metals of the vanadium and chromium group, cannot be obtained from classical Morse or Lennard-Jones type potentials, but needs the range flexibility within the pair potential. View Full-Text
Keywords: crystal-lattice symmetry; Platonic Solids; elemental crystal structures; pair potential; atomic interaction range; phase diagram crystal-lattice symmetry; Platonic Solids; elemental crystal structures; pair potential; atomic interaction range; phase diagram
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This is an open access article distributed under the Creative Commons Attribution License which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. (CC BY 4.0).
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Zschornak, M.; Leisegang, T.; Meutzner, F.; Stöcker, H.; Lemser, T.; Tauscher, T.; Funke, C.; Cherkouk, C.; Meyer, D.C. Harmonic Principles of Elemental Crystals—From Atomic Interaction to Fundamental Symmetry. Symmetry 2018, 10, 228.

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