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Average Structure Of Quasicrystals José Luis Aragón Vera Centro de Física Aplicada y Tecnología Avanzada, Universidad Nacional Autónoma de México. Gerardo G. Naumis Instituto de Física, Universidad Nacional Autónoma de México. Rafael A. Barrio Instituto de Física, UNAM, México D.F., México Manuel Torres Instituto Superior de Investigaciones Científicas Madrid, España. Michael Thorpe Arizona State University, Tempe, Arizona, USA.

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Summary The main problem: since quasicrystals lack periodicity, conventional Bloch theory does not apply (electronic and phonon propagation) Average structure in 1D and 2D Conclusions.

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Quasicrystal A material with sharp diffraction peaks with a forbidden symmetry by crystallography. They have long-range positional order without periodic translational symmetry

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Quasicrystals as projections x ( ) Since the star is eutactic, there exists an orthonormal basis {e 1,e 2,...,e 5 } in R 5 and a projector P such that P(e i )=a i, i=1,..,

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E EE The cut and projection method in R 2 22

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(E (E The reciprocal space of a quasicrystal 2*2*

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Bloch’s theorem The eigenstates of the one-electron Hamiltonian H= ~ 2 r 2 /2m + U(r), where U(r+R)=U(r) for all R in a Bravais lattice, can be chosen to have the form of a plane wave times a function with the periodicity of the lattice: where for all R in the Bravais lattice.

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In a crystal: a Bragg spot in the diffraction pattern can open a gap in the electronic density of states since the wave is diffracted (with such a wave-length, it has the same “periodicity” of the lattice and becomes a standing wave). The reciprocal space of a quasicrystal is filled in a dense way with Bragg peaks Thus, the density of states is full of singularities (1D), (2D and 3D??) Van Hove singularities Since quasicrystals lack periodicity, conventional Bloch theory is not useful. There are however indications that Bloch theory may be applicable in quasiperiodic systems: G k E(k)

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3. Albeit the reciprocal space of quasicrystal is a countable dense set, it has been shown that only very few of the reciprocal-lattice vectors are of importance in altering the overall electronic structure. A.P. Smith and N.W. Ashcroft, PRL 59 (1987) To a given quasiperiodic structure we can associate an average structure whose reciprocal is discrete and contains a significant fraction of the scattered intensity of the quasiperiodic structure. J.L. Aragón, Gerardo G. Naumis and M. Torres, Acta Cryst. A 58 (2002) Through angle-resolved photoemission on decagonal Al 71.8 Ni 14.8 Co 13.4 it was found that s-p and d states exhibit band-like behavior with the rotational symmetry of the quasiperiodic lattice. E. Rotenberg et al. NATURE 406 (2000) 602.

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A classical experiment Liquid: Fluorinert FC75 Tiling edge length8 mm Number of vertices (wells)121 Radius of cylindrical wells1.75 mm Depth of cylindrical wells2 mm Liquid depth0.4 mm Frequency35 Hz

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Snapshots of transverse waves 0.04 s 0.08 s 0.00 s 0.24 s Wave pulse is launched along this direction

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The quasiperiodic grid The above quasiperiodic sequence (silver or octonacci) can be generated starting from two steps L and S by iteration of substitution rules: L ! LSLS ! L.

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-X Testing the Bloch-like nature The quasiperiodically spaced standing waves can be consi- dered quasiperiodic Bloch-like waves if they are generated by discrete Bragg resonances. 1. The quasiperiodic sequence:

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Average structure of a quasicrystal For phonons: what is the sound velocity? Dynamical structure factor? where the Green´s function is given by,

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For a Fibonacci chain the positions are given by: but: S L L S L 12 3

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Sound velocity : G.G. Naumis, Ch. Wang, M.F. Thorpe, R.A. Barrio, Phys. Rev. B59, (1999)

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The generalized dual method (GDM) Star vector3-Grid

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The generalized dual method (GDM) 1. Select a star-vector: Each region can be indexed by N integers defined by its ordinal position in the grid. 2. The equations of the grid are where n j is an integer, x 2 R 2 and j are shifts of the grid with respect to the origin. (2,2,-1)

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For the direction e l, the ordinal coordinates are: where e j ? is perpendicular to e j and a jk is the area of the rhombus generated by e j and e k. 3. Finally, the dual transformation associates to each region the point t is then a vertex of the tiling.

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A formula for the quasilattice By considering the pairs (jk), we obtain a formula for the vertex coordinates of a quasiperiodic lattice: Gerardo G. Naumis and J.L. Aragón, Z. Kristallogr. 218 (2003) 397. where, and.

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The average structure is a fluctuation part that we expect to have zero average in the sense that By using the identity, the equation for the quasilattice can be written as defines an average structure which consists of a superposition of lattices.

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Multiplicity=4 Multiplicity=2

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Properties of the average lattice 1. The reciprocal of the average structure contains a significant fraction of the scattered intensity of the quasiperiodic structure. 2. The average structure dominates the response for long- wave modes of incident radiation. 3. The average structure then can be useful to determine the main terms that contribute to define a physically relevant Brillouin zone. J.L. Aragón, Gerardo G. Naumis and M. Torres, Acta Crystallogr. A 58 (2002) 352.

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