Interfaces and quasicrystals as competing crystal lattices: Towards a crystallographic theory of interfaces

Abstract

This paper shows that in a higher-dimensional approach, quasicrystals and interfaces are formally equivalent. Interfaces and quasicrystals are interpreted here as a region in space where the atoms of interpenetrated crystal lattices compete for space. Based on this paradigm, a method derived from the strip-projection method developed for the study of quasicrystals has been introduced. The method is completely general, independent of the parent crystal lattice type, relative orientation, and translation and of the position and orientation of the boundary plane. In this approach the perpendicular space coincides with Bollmanns' displacement space, while the parallel space contains a physical structure characterized by a minimum local strain that includes both the interface and adjacent crystal lattices. A classification of interfaces in a finite number of well-defined equivalence classes ͑local isomorphisms͒ that include orientational and translational degrees of freedom has been introduced. This classification is based on the symmetry of the hyperlattice and the position and shape of the strip and incorporates concepts from previous structural units and symmetry breaking approaches. It is suggested that such classes can be related to physical properties of interfaces. The formalism defines ideal ͑minimum strain͒ structures assumed to play an analogous role in grain boundaries ͑GB's͒ to those played by the perfect crystal and quasicrystal concepts in the study of crystals and quasiperiodic structures. Also, a lattice, called the phason lattice, is introduced to account for the dislocation content of nonsingular interfaces. Accordingly, the properties of any GB are seen to be determined by the periodicity of isosymmetrical regions related to the O and phason lattices and not by the ill-defined and pathologically discontinuous index number ⌺.

Symmetry classification of cubic twist grain boundaries

“…Translational states are structures of different symmetry obtained by relative translations of L 1 and L 2 . In a higher-dimensional description, such translations are equivalent to a displacement of the strip [1].…”

“…The hyperlattice is obtained by embedding two 3D lattices generated by the structure matrices L 1 , L 2 in 6D space [1], its structure matrix is given by…”

“…The purpose of this paper is to present a new classification of interfaces derived from a higherdimensional approach to the study of interfaces [1]. This classification, meant to serve as a step towards the creation of a crystallography of interfaces, groups GBs in equivalence classes defined by their symmetry, and it is expected that the properties of interfaces could be subsequently related to the equivalence class they belong to.…”

Modeling Grain Boundaries in Solids Using a Combined Nonlinear and Geometrical Method

“…7. Briefly, the crystal lattices are embedded in a higher dimensional space where they define an hyper lattice whose symmetry depends on that of the lattices and their relative orientation.…”

“…7 by the use of quasi-crystal methods. We briefly recall the main aspects of this second theory below.…”