Dislocations in a penrose lattice

Abstract

We have made a general analysis of the nature of a dislocation in a quasicrystal, which we define as the intersection of a Volterra dislocations in a ddimensional hypercubic lattice with “physical” space P ✓ (for a 3-dimensional quasicrystal, d = 6, d ✓ = 3; for a Penrose lattice, d = 5, d ✓ = 2 ). We show that the dislocation hyperline in d-space is a ( d-2)-dimensional manifold which for stability reasons is most probably the direct product of the dislocation line (or point) L ✓ in d ✓- space with the “perpendicular” space P ⊥. We have simulated the elastic field of deformation and the so-called “phason” field around certain dislocations in a Penrose lattice (A) for usual Burgers' vectors b ✓ as well as very small b ✓s (by virtue of the irrationality, b ✓ and b ⊥ can take any value, only the sum b = b ⊥ + b ✓ is restricted to a translation vector of the hypercubic lattice), and (B) for various positions corresponding to glide and climb motions. We have shown that glide and climb concepts in a quasicristal are equivalent to the same concepts in a usual crystal and have suggested that the rearrangement of “phasons” presents some analogies with the process of reshuffling in some close-packed oxydes. However the question of the core remains of intriguing interest.