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PHILOSOPHICAL MAGAZINE A, 1999, VOL. 79, NO. 3, 549560
Determination of the tiling type and phason strain analysis
of decagonal quasicrystals in AlNiCo alloys
By R. Hory, C. Pohla and P. L. Ryder
Institut f u r Werstophysik und Strukturforschung, University of
Bremen,Germany
[Received 29 January 1998 and accepted in revised form 7 May
1998]
Abstract
From high-resolution electron micrographs of the decagonal
phases of theAl72.5Co11 Ni16 .5 and Al70Co15 Ni15 alloys, tilings
are produced by connectingprominent structural features. Indexing
the tiling vertices as the projection of ave-dimensional periodic
lattice and investigation of the distribution of thevertices in
orthogonal space gives information about the tiling model
andpossible phason strains. The technique also reveals domain
boundaries whichseparate regions diering only in the amounts of
linear phason strain.
1. IntroductionThe structures of quasicrystals can be regarded
as the intersection of a periodic
structure in 3 n dimensions n 1,2,3 with the three-dimensional
physical spaceE (Janssen 1988). The n-dimensional subspace is
called the internal or orthogonalspace E . The well known Penrose
tilings and their generalizations (Penrose 1974,
Janssen 1988) are frequently used models for quasicrystal
lattices.Elastic excitations in the physical and orthogonal spaces
correspond to phonons
and phasons respectively (Lubensky et al. 1985). The relaxation
of a phason eld isassociated with the diusion of atoms. Thus phason
defects can easily be quenched
into quasicrystals and are dicult to remove by annealing. In the
tiling modelsof quasicrystalline structures, phasons correspond to
local rearrangement of the
vertices, so-called phason ips (Elser 1985, Tang and Jaric
1990).The structural changes caused by phasons inuence the
diraction properties of
quasicrystals. A linear phason strain, for example, results in
the displacement of the
Bragg spots in electron diraction patterns and a change in the
proles of X-ray
reections (Lubensky et al. 1986). A phason strain whose Fourier
components areindependent random variables is called a random
phason strain and leads to a
broadening or even disappearance of Bragg peaks (Lubensky et al.
1986). Otherphysical properties, such as the electrical
conductivity or the specic heat, are alsoinuenced by a phason
strain (Yamamoto and Fujiwara 1995, Wang and Garoche
1997). In addition, certain types of linear phason strain may
transform the perfectquasicrystal into a crystalline approximant.
Such transformations have been
observed in various alloy systems (Hu and Ryder 1994, Zhang and
Kuo 1990,
Cheng et al. 1992).The phason strain tensor can be calculated in
principle from the displacements of
the spots in the electron diraction patterns or from the X-ray
diraction
peak proles ( Lubensky et al. 1986, Edagawa 1990). However,
these methods, espe-cially the latter, provide only a mean value
over a certain volume of the quasicrystal.
01418610/99 $12.00 1999 Taylor & Francis Ltd.
8/3/2019 R. Hory, C. Pohla and P. L. Ryder- Determination of the
tiling type and phason strain analysis of decagonal quasicr
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High-resolution transmission electron microscopy (HRTEM) allows
local variationsin the phason strain to be detected on a very small
scale (Li et al. 1992, Jiang and
Kuo 1994).In the electron microscope image of a decagonal
structure taken along the dec-
agonal axis, characteristic ring features are visible,
corresponding to the local dec-
agonal arrangement of columns (Hiraga et al. 1991, Beeli and
Horiuchi 1994 ). Inorder to reveal the tiling which is
characteristic of the particular quasicrystal, the ring
centres must be connected by straight lines with a xed base
length a0. In the presentpaper, the tiling determined in this way
is used to measure the local phason strain
matrix, to detect domains with dierent phason strains and to
identify the type of
tiling by analysing the distribution of the vertices in E .
2. Theory
2.1. Indexing
When the tiling has been produced as described above, the
vertices can be
indexed and `lif ted into the ve-dimensional space, for example
by a technique
rst proposed by Chen et al. (1990) and He et al. (1991). This
procedure makesuse of the fact that two-dimensional quasiperiodic
tilings can be represented as the
projection of a periodic ve-dimensional lattice.
Let E5
be the ve-dimensional hyperspace and ei the base vectors of the
lattice,
with
ei2
5
1 /2
cos2p i 1
5, sin
2p i 1
5, cos
4p i 1
5,
sin4p i 1
5,
1
21 /2,
E the two-dimensional physical space spanned by the base
vectors
ei2
5
1 /2
cos2p i 1
5, sin
2p i 1
5,
and E the three-dimensional orthogonal space spanned by the
vectors
ei2
5
1 /2
cos4p i 1
5, sin
4p i 1
5,
1
21 /2,
where i 1, . . . ,5.The projection direction is the hyperlattice
unit cell diagonal d 1,1,1,1,1 .A vertex x in E can thus be
assigned the ve indices n1, . . . ,n5 such that
x
5
i 1
niei.
With the aid of these indices the projection in E may be
calculated.
x
5
i 1
niei .
550 R . Hory et al.
