J. Phys. I France
Volume 2, Numéro 1, January 1992
Page(s) 69 - 87
DOI: 10.1051/jp1:1992124
J. Phys. I France 2 (1992) 69-87

Dislocations and disvections in aperiodic crystals

Maurice Kléman

Laboratoire de Physique des Solides (Associé au C.N.R.S.), Université de Paris-Sud, Bât. 510, 91405 Orsay Cedex, France

(Received 25 July 1991, accepted 4 October 1991)

The search for topological invariants of dislocations in a periodic crystal employs mappings of the Burgers circuit surrounding the defect on 2 types of manifolds: the perfect lattice (this mapping yields the Burgers vector) and the order parameter space (which classifies the defects as elements of the homotopy groups of this op space). Both classifications are equivalent because the perfect lattice is a covering (in a topological sense) of the op space. We define an aperiodic crystal as the $d_{\parallel}$-dimensional boundary of a d-dimensional crystal and get two similar manifolds  : an op space $\mathbb{U} $ which is the acceptance domain in $\mathbb{P} _\perp$ with suitable identifications of the faces, and a perfect lattice $\mathbb{H} ^d$ which is a curved periodic lattice in a space of negative curvature and a covering of the former. $\mathbb{H} ^d$ is invariant under the action of a non-abelian group of translations H d which classifies all the ( $d_\perp
-2$ )-dimensional singularities of the aperiodic crystal, viz. a) complete dislocations and b) novel topological defects which we call disvections. The latter are classified as the elements of some normal subgroup $\Sigma_{\rm a}$ of H d which includes the commutator subgroup. The former appear as the elements of the quotient group of H d mod. $\Sigma_{\rm a}$ or else as the elements of a subgroup of the group of dislocations Zd of the hypercubic lattice; the other elements of Zd which do not enter in H d correspond to partial dislocations. We argue that disvections are the usual phason defects (mismatches, etc.). The above theory of defects applies, mutatis mutandis, to approximants, for which we provide a topological classification.

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