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.
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.
Quasicrystal
A material with sharp diffraction peaks with a forbidden symmetry by crystallography.
They have long-range positional order without periodic translational symmetry
V ( x)  cos( x)  cos( x),   p / q
V ( x)  V ( x  T )
V ( x)   V (q ) cos( qx)
q
1
V (q)   (q  n)   (q  ( )m) 
2
Quasicrystals as projections
(
5
1
1
1
0
-1 1
)
x
4
2
3
Since the star is eutactic, there
exists an orthonormal basis
{e1,e2,...,e5} in R5 and a projector
P such that P(ei)=ai , i=1,..,5.
The cut and projection method in
2
E||
E^
e2
e1
2
R
The reciprocal space of a quasicrystal
V  F (W (r )r )  F (W (r ))  F (r )
2*
(E||*
(E^*
*
2
e
*
1
e
Bloch’s theorem
The eigenstates  of the one-electron Hamiltonian
H=~2r2/2m + 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.
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
E(k)
k E (k )  0
G
k
dS
 (E)  
3
4

 k E (k )
Sn ( E )
There are however indications that Bloch theory may be
applicable in quasiperiodic systems:
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) 1365.
4. 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.
5. 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.
A classical experiment
Liquid: Fluorinert FC75
Tiling edge length
Number of vertices (wells)
Radius of cylindrical wells
Depth of cylindrical wells
Liquid depth
Frequency
8 mm
121
1.75 mm
2 mm
0.4 mm
35 Hz
Snapshots of transverse waves
0.00 s
0.04 s
Wave0.24
pulsesis launched
along this direction
0.08 s
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.
Testing the Bloch-like nature
The quasiperiodically spaced standing waves can be considered quasiperiodic Bloch-like waves if they are generated
by discrete Bragg resonances.
1. The quasiperiodic sequence:
G-X
Average structure of a quasicrystal
For phonons: what is the sound velocity?
1
S ( q,  )  
N
2
e
iq ( xi  x j )
Dynamical structure factor?
ImGij ( )
2
i, j
where the Green´s function is given by,
ul (i )ul ( j )
Gij ( )   2
2



l 1
l
2
( H   2 )G( 2 )  1
J ij
 pi2
2
H  
  (ui  u j ) 
i  2mi
j 2

For a Fibonacci chain the positions are given by:
but:
n
xn  nS  ( L  S )  
 
x   x   x
n
xn  n   ( L  S )  
 

1 5
2

S
L
L
S
L
x
1
S (q,  2 ) 
N

e
N

3
iq  ( n  m ) iq ( L  S )({n /  }{m /  })
e
n , m 1

2
e
iq  ( n  m )
Im Gn,m ( 2 )
(1  iq( L  S )({n /  }  {m /  }) 
n , m 1
q 2 ( L  S )({n /  }  {m /  }) 2  ...) Im Gn,m ( 2 )
2
2

(



(q   2 n /  ))
q
(
L

S
)


2
2
S (q,  )  4sin 

2
(q( L  S )  2 n) 2

 n 0
Sound velocity:
vs  J 
G.G. Naumis, Ch. Wang, M.F. Thorpe, R.A. Barrio, Phys. Rev. B59, 14302 (1999)
The generalized dual method (GDM)
3-Grid
Star vector
The generalized dual method (GDM)
4
-2
-3
-1
0
1
2
3
4
1. Select a star-vector:
3
2
1
0
-1
-2
-3
-4 -3 -2 -1 0 1 2 3 4
3
0
2
-1
2
3
2. The equations of the grid are
where nj is an integer, x 2 R2 and j
are shifts of the grid with respect to
the origin.
Each region can be indexed by N
integers defined by its ordinal
position in the grid.
(2,2,-1)
For the direction el, the ordinal coordinates
are:
where
ej? is perpendicular to ej and ajk is the area
of the rhombus generated by ej and ek.
3. Finally, the dual transformation associates
to each region the point
t is then a vertex of the tiling.
A formula for the quasilattice
By considering the
pairs (jk), we obtain a formula for
the vertex coordinates of a quasiperiodic lattice:
where
,
and
.
Gerardo G. Naumis and J.L. Aragón, Z. Kristallogr. 218 (2003) 397.
The average structure
By using the identity
quasilattice can be written as
, the equation for the
defines an average structure which
consists of a superposition of
lattices.
is a fluctuation part that we expect to
have zero average in the sense that
Multiplicity=4
Multiplicity=2
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 longwave 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.