R. Hory, C. Pohla and P. L. Ryder- Determination of the tiling type and phason strain analysis of...
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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.
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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 .
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Since5i 1 ei 0 (and
5i 1 ei 0 , the indexing is not unique without a
further restriction, which we shall arbitrarily take to be
0