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

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.

Three-dimensional lattice ground states for Riesz and Lennard-Jones type energies

“…This is new compared to the two-dimensional case where the square lattice that have been shown in [4] to be locally minimal in some interval of volume is also observed to be minimal in L 2 (V ) in the same interval. Furthermore, our numerical findings concerning the global minimizer of E LJ n,m in L 3 support [5] and confirm [2].…”

“…Using dimension reduction techniques developed in [12] for Lennard-Jones type energies, we investigate the minimum of E LJ n,m in L 3 . As already explained in [2], only two lattices appear to be global minimizers of this energy: the FCC lattice and the HCP structure (see Figure 5). It seems that a large (resp.…”

“…In dimension d = 2, numerical investigations suggest that the ground state of the classical Lennard-Jones energy (i.e. (n, m) = (12,6)) is a triangular lattice [15], whereas the Hexagonal Closed-Packing structure is expected to be the one in dimension d = 3 [2]. Notice also that the ground states of three-dimensional finite-range Lennard-Jones energy among a finite number of structures has been also studied in [52] for different truncation, cutoff distances, pressure and exponents.…”

Optimality of the triangular lattice for Lennard-Jones type lattice energies: a computer-assisted method

“…In dimension 3, the same is expected to happen on a Hexagonal Close Packing structure (see e.g. [1,4]). The rare attempts to show the two-dimensional case have lead to optimality results for the triangular lattice in the case of hard-sphere potentials [24,30] and approximations of them [23,33,39].…”

Lattice ground states for embedded-atom models in 2D and 3D

“…• As mentioned before, the unique minimizer of the Lennard-Jones energy E φ in L 3 is λ φ D 3 D 3 (see, e.g., [6] and [39,Fig. 5]).…”