Structures of quasicrystals: Where are the atoms?

Abstract

This paper is an attempt to present a chronological review of the structural concepts that have been developed to characterize quasicrystals, starting from the experimental discovery of D. Shechtman and the concomitant theoretical definition of quasicrystal as proposed by D. Levine and P. Steinhardt, up to the present research in the field. The largest part of the paper is a discussion of the specific points that make the atomic structure determination of quasicrystals an original and difficult scientific challenge. We finally discuss the soundness of our knowledge of the actual atomic structure in quasicrystals: we do have quite a solid idea of where the atoms are but we are not sure about the distribution of the chemical species.

Keywords:

• crystallography
• quasicrystals

Notes

Notes

1. A dense set of peaks such that peaks with an intensity greater than any strictly positive threshold are discrete.

2. A ℤ-module is a vector space built on a ring; it can equivalently be seen as the projection in a space of dimension d along an irrational direction of a regular periodic lattice embedded in a space of dimension N > d; N is the rank of the ℤ-module.

3. When the N-dimensional lattice projects uniformly densely in E _⊥, any finite patch of tiles which appears in a given tiling appears in any other tiling defined by a cut parallel to the initial one and with the same frequency.

4. This requirement is not fulfilled in incommensurate structures and could be one of the distingo between quasicrystals and incommensurate phases.

5. We say that a tiling has matching rules if there exists a set of local constraints that enforce quasiperiodicity of the tiling as long as they are satisfied everywhere in the tiling.

6. To allowing an easy bulk reconstruction, the cut should easily ‘glide’ in the 6D space along E _⊥ in exploring the class of the indistinguishable structures through low-energy atomic jumps with no transport of matter at long distances.

7. Even if E _σ is rationally oriented, and generates a quasiperiodic decoration of a usual (periodic) lattice, the carrier of the Fourier transform is a non-crystallographic ℤ-module. The fact that the Fourier spectrum can have a carrier that is not based on the reciprocal lattice of the host lattice is because the quasiperiodic decoration is not the convolution of a finite pattern with the host lattice.

8. Expressing the coordinates of a reflection Q in the internal basis of as Q = κ(h + h′τ, k + k′τ, ℓ + ℓ′τ), with , h, h′, k, k′, ℓ, ℓ′ ∈ ℤ, leads us to define the two numbers N = h ² + h′² + k ² + k′² + ℓ² + ℓ′² and M = h′² + k′² + ℓ′² + 2(hh′ + kk′ + ℓℓ′) that define the length of the reflection Q in and : |Q _∥|² = κ²(N + Mτ) and |Q _⊥|² = κ²τ(Nτ − M).

9. To avoid confusion between the two bc positions, the occupied one has the same parity as the smallest of the two ASs of the nodes.

10. This technique is better for characterizing the position of the projected 6D nodes in the 2D-plane with respect to two simple vectors of that plane.

11. In both cases, the complementary 4D space, say P _U , is defined by the projector