Average Structure Of Quasicrystals

1
Average Structure Of Quasicrystals
Gerardo G. Naumis Instituto de Física,
Universidad Nacional Autónoma de México.

• José Luis Aragón Vera
• Centro de Física Aplicada y Tecnología Avanzada,
  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.
2
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.

3
Quasicrystal
A material with sharp diffraction peaks with a
forbidden symmetry by crystallography. They have
long-range positional order without periodic
translational symmetry
4
Quasicrystals as projections
( )
1
1
0
-1
1
Since the star is eutactic, there exists an
orthonormal basis e1,e2,...,e5 in R5 and a
projector P such that P(ei)ai , i1,..,5.
5
The cut and projection method in R2
E
E
6
The reciprocal space of a quasicrystal
?2
(E)
(E)
7
Blochs theorem
8
Since quasicrystals lack periodicity,
conventional Bloch theory is not useful.

• 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

There are however indications that Bloch
theory may be applicable in quasiperiodic systems
9

1. 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) 1365.
2. 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) 352.
3. Through angle-resolved photoemission on decagonal
   Al71.8Ni14.8Co13.4 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.

10
A classical experiment
Liquid Fluorinert FC75 Tiling edge length 8 mm
Number of vertices (wells) 121 Radius of
cylindrical wells 1.75 mm Depth of cylindrical
wells 2 mm Liquid depth 0.4 mm
Frequency 35 Hz
11
Snapshots of transverse waves
0.24 s
12
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 ! LSL S ! L.
13
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
G-X
14
Average structure of a quasicrystal
For phonons what is the sound velocity?
Dynamical structure factor?
where the Greens function is given by,
15
(No Transcript)
16
For a Fibonacci chain the positions are given by
but
S
L
L
L
1
2
3
17
Sound velocity
G.G. Naumis, Ch. Wang, M.F. Thorpe, R.A. Barrio,
Phys. Rev. B59, 14302 (1999)
18
The generalized dual method (GDM)
19
The generalized dual method (GDM)
Each region can be indexed by N integers defined
by its ordinal position in the grid.
(2,2,-1)
20
(No Transcript)
21
A formula for the quasilattice
Gerardo G. Naumis and J.L. Aragón, Z.
Kristallogr. 218 (2003) 397.
22
The average structure
23
Multiplicity4
Multiplicity2
24
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.