Phonon-Phason coupling in icosahedral quasicrystals

– From relaxation simulations of decoration-based quasicrystal structure models using microscopically based interatomic pair potentials, we have calculated the (usually neglected) phonon-phason coupling constant. Its sign is opposite for the two alloys studied, i-AlMn and i-(Al,Cu)Li; a dimensionless measure of its magnitude relative to the phonon and phason elastic constants is of order 1/10, suggesting its effects are small but detectable. We also give a criterion for when phonon-phason effects are noticeable in diffuse tails of Bragg peaks. The elastic description of an icosahedral quasicrystal includes both “phonon” (ordinary) strain and “phason” strain. “Phason” strain parametrizes the way in which the local tile distribution deviates from icosahedral symmetry, as forced by the constraint of packing with surrounding tiles. Correspondingly there are two easily measured “phonon” elastic constants (ordinary bulk and shear modulus), and two “phason” elastic constants. The latter have been computed theoretically for various tilings [1, 2, 3, 4] and recently measured experimentally [5]. There remains a cross-term with one elastic constant, the “phonon-phason” coupling Γ, which has not previously been measured experimentally or theoretically. The constant Γ enters the shape of the diffuse tails around Bragg peaks [6], can drive a quasicrystal phase unstable with respect to a related crystal phase [7, 8], and affects the strain fields of dislocations in quasicrystals, which have been studied by diffraction contrast in transmission electron microscopy [9, 10]. Previous authors [5, 6, 10, 11] simply assumed Γ was small compared with the other elastic constants. The present work is the first attempt to compute Γ theoretically. As explained below, we extract it numerically from simulations in which we relax the atomic positions of a finite “periodic approximant” of the quasicrystal and measure its spontaneous shear distortion in response to the its intrinsic phason strain, for several different approximants. This was carried out for moderately realistic models of both i-AlMn and i-(Al,Cu)Li (modeled as a () Present address: IBM T.J. Watson Research Center P.O. Box 218, Yorktown Heights, NY 10598 Typeset using EURO-TEX 2 EUROPHYSICS LETTERS pseudo-binary), representing the two major classes of quasicrystal. (Here “model” denotes a combination of a tiling, an atomic decoration rule, and a microscopically based pair potential.) In our conclusion, we crudely estimate ratios which determine the experimental observability of Γ. Elastic theory. – This is formulated in terms of two kinds of strain fields: the “phonon” (ordinary) strain tensor has components uij = (∂iuj + ∂jui)/2 where u(r) is the phonon displacement. The phason strain components are vij = ∂ivj . Here v(r) is the phason (also called “perp”) displacement field, defined for quasicrystals and/or tilings; it can be constructed for any configuration of the tiles [12, 13], and parameterizes the local imbalance in tiles of different orientations. Periodic approximant structures [15] of the quasicrystalline state necessarily acquire a non-zero phason strain. Group theory [6, 14] permits the following terms in the elastic free energy (in the notation of [7], and adopting the summation convention): F = Fphonon + Fphason + Fphonon−phason (1)