Tiling models for metadislocations in AlPdMn approximants
7 Tiling models for metadislocations in AlPdMn
approximants
M. Engel∗ and H.-R. Trebin
Institut für Theoretische und Angewandte Physik, Universität Stuttgart,
Pfaffenwaldring 57, D-70550 Stuttgart, Germany
November 28, 2018
Abstract
The AlPdMn quasicrystal approximants ξ, ξ′, and ξ′n of the 1.6 nm decagonal
phase and R, T , and Tn of the 1.2 nm decagonal phase can be viewed as arrange-
ments of cluster columns on two-dimensional tilings. We substitute the tiles by
Penrose rhombs and show, that alternative tilings can be constructed by a simple
cut and projection formalism in three dimensional hyperspace. It follows that in the
approximants there is a phasonic degree of freedom, whose excitation results in the
reshuffling of the clusters. We apply the tiling model for metadislocations, which are
special textures of partial dislocations.
1 Introduction
A quasiperiodic structure can be described as a cut through a hyperlattice decorated with
atomic surfaces. As a consequence the atoms are arranged in a finite number of different
local environments frequently leading to a substructure of highly symmetric clusters. The
cluster positions can again be modelled by a simpler decoration consisting in the simplest
case of exactly one atomic surface (for the cluster centre) per hyperlattice unit cell. The
displacement of the cut space (phasonic displacement) is a discrete degree of freedom,
called phasonic degree of freedom. It can be excited locally, leading to a rearrangement of
the clusters by correlated atomic jumps. This view is supported by recent diffraction data
of coherent phason modes in i(cosahedral)-AlPdMn (Coddens et al. 1999, Francoual et al.
2003) and by in situ observations of phason jumps via high-resolution transmission electron
microscopy (HRTEM) in d(ecagonal)-AlCuCo (Edagawa et al. 2002).
The cut formalism cannot be applied directly if there is a gradient in the phasonic
displacement. This is examplified by the extreme case of a cut space running inbetween
∗Author for correspondence. Email: mengel@itap.physik.uni-stuttgart.de
http://arxiv.org/abs/0704.1440v1
all atomic surfaces and not touching any of them. Hence we resort to another method:
We substitute the atomic surfaces by atomic hypervolumes (Engel and Trebin 2005) of the
same dimension as the hyperspace. For the construction two different spaces are needed:
Those atomic hypervolumes that are cut by the (possibly deformed) cut space ECut are
selected. The centre of each selected atomic hypervolume is projected onto the projection
space EProj. By shearing the cut space, i.e. by introducing a linear phasonic displacement,
periodic approximants are created. Phasonic degrees of freedom also can exist in these and
play a fundamental role for phase transitions (Edagawa et al. 2004).
Here we discuss linear defects, metadislocations, in the phasonic degree of freedom for
approximants of the AlPdMn system. This system is especially adequate for the exam-
ination of phasonic degrees of freedom since a stable i-phase, a stable 1.2 nm d-phase,
a metastable 1.6 nm d-phase (which is assumed to be a solid solution of Mn in d-AlPd
(Steurer 2004)), and a large variety of approximants have been observed in the phase
diagramm (Klein et al. 2000). All of them, as well as several binary AlPd and AlMn
quasicrystals and approximants are related structurally.
2 Tiling models
A hyperspace model for i-AlPdMn has been proposed by Katz and Gratias (1993). It uses
a six dimensional face-centred hyperlattice F 6D with lattice constant 2l6D = 1.29 nm and
serves as a starting point since to our knowledge all newer, more complicated hyperspace
models are refinements. It has been shown, that the approximants ξ and ξ′ (Beraha et al.
1997) of the 1.6 nm d-phase and the approximants R and T (Beraha et al. 1998) of the
1.2 nm d-phase are described on an atomistic level by a shear in this model. However the
authors had to introduce an additional mirror symmetry to assure full tenfold symmetry.
The main building units are Mackay-type clusters, whose centres are projected from the
hyperspace by using one atomic surface per hyperlattice unit cell. It is a subset of a
triacontahedron, deflated by τ = 1
5+1) with respect to the canonical triacontahedron,
which is the projection of the hypercube with edge length l6D on the orthogonal complement
of EProj. The distance of the cluster centres is the shortest projection of neighboring F
sites ei ± ej multiplied by the deflation factor: t6D = 15
τ + 2l6D ≃ 0.78 nm.
The relation of the approximants to the i-phase is given by two consecutive shears of
ECut in the hyperspace: The first shear changes the cluster arrangement in direction of a
fivefold axis e1. Together with the introduction of a mirror plane this results in a decagonal
quasicrystal. The clusters are then aligned in columns parallel to the tenfold axis, so that
the structures can be described by two-dimensional tilings, which are the projections in
the column direction. The second shear rearranges the columns perpendicular to e1. We
will now consider the tilings for the 1.6 nm and the 1.2 nm phases separately.
Neighboring clusters in the 1.6 nm d-phase and its approximants lie on planes per-
pendicular to e1. Thus the tile length is t
6D as defined above. The ξ- and the ξ′-phase
are built from flattened hexagons (H) arranged in parallel and in alternated orientation
respectively. By introducing additional rows of pentagons (P) and nonagons (N) between
PSfrag replacements
1.6 nm d-phase
1.2 nm d-phase
D U N H P
D’ S’ B’ H’
AR AR’OR OR’
Figure 1: Calculated tilings for various approximants of the AlPdMn i-phase. (a) 1.6
nm phases: The tiling of the d-phase is the Tübingen Triangle Tiling (TTT). (b) 1.2 nm
phases: The D’ centers lie on a τ 2 inflated TTT. (c) The tiles can be substituted with
Penrose rhomb tiles. The substitution is different for the 1.6 nm phases and the 1.2 nm
phases.
each n− 1 rows of alternated Hs in the ξ′-phase we obtain the ξ′n-phases. Such a PN-row
is called a phason plane, which is justified since it is elongated in the e1 direction. As
we will see, phason planes play an important role for the cluster reshuffling resulting from
an excitation of the phasonic degree of freedom. Further tiles, the decagon (D) and an
U-shaped tile (U) are observed in the 1.6 nm d-phase (Fig. 1 (a)).
Neighboring clusters in the 1.2 nm d-phase and its approximants lie on two planes
staggered perpendicular to e1 with distance
10l6D ≃ 0.41 nm. The resulting tile length
is t′6D = 1
10τl6D ≃ 0.66 nm. The R- and the T -phase are built from elongated hexagons
(H’) arranged in parallel and in alternated orientation respectively. The Tn-phases are
created by introducing into the T -phase additional rows of boat-shaped tiles (B’) between
each n rows of alternated H’s. A B’-row is again called a phason-plane. For the 1.2 nm
d-phase additionally a decagon (D’) and a star-shaped tile (S’) are needed (Fig. 1 (b)).
By substituting the H, P, N, H’, and B’ tiles with acute rhombs (AR), (AR’) and obtuse
rhombs (OR), (OR’) as shown in Fig. 1 (c) we obtain new tilings for the approximants,
which can be interpreted as approximants of the Penrose-tiling. An H is substituted by
an OR, while a phason plane corresponds to a combination of an AR-row and an OR-
row. So the ξ′n-phase has n OR-rows inbetween neighboring AR-rows. Similarly the Tn-
phase has n AR-rows inbetween neighboring phason planes, represented by OR-rows. The
rhombs occuring in the new tilings for the Ξ-approximants (ξ, ξ′, ξ′n), as well as for the T -
approximants (R, T , Tn) both only need three of the five basis vectors of the Penrose-tiling
to be constructed. Therefore the tilings can be modelled in a simple three-dimensional
hyperspace with the Z3-lattice and lattice constant l3D = τ
τ + 2l6D ≃ 1.99 nm. The
projection matrices are (si = sin(2π
), ci = cos(2π
s0 s1 s4
c0 c1 c4
s0 s2 s3
c0 c2 c3
, (1)
leading to an edge length of the tiles: t3D = 1
10l3D ≃ 1.26 nm. There is one atomic
hypervolume per unit cell, which is just the unit cell, and one phasonic degree of freedom.
This three-dimensional hyperspace is the simplest model for a phasonic degree freedom
besides the Fibonacci-chain.
3 Metadislocations
In the formalism of atomic hypervolumes a dislocation can be introduced into a tiling by
a generalised Voltera process (Engel and Trebin 2005). It is uniquely characterised by a
translation vector of the hyperlattice, the Burgers vector b (here: b3D = (b1, b2, b3), bi ∈ Z),
that splits up into a phononic component b‖ = π‖b (deforming the tiles) and a phasonic
component b⊥ (rearranging the tiles). The latter can only be calculated from the full six-
dimensional Katz-Gratias model. If it is not zero, such a dislocation is a partial dislocation.
By (i) extending the linear theory of elasticity to the hyperspace, (ii) approximating the
phasonic degree as continuous, and (iii) assuming isotropy in the strain fields, the line
PSfrag replacements
1.6 nm d-phase
1.2 nm d-phase
Figure 2: Tilings of the ξ′3- (left) and the T3-phase (right) with m = 4 metadislocations.
2Fm new phason planes are inserted from the left, ending at the triangular shaped dislo-
cation core.
energy E of a dislocation is expressed as:
E = cphon‖b‖‖2 + cphas‖b⊥‖2 + ccoupl‖b‖‖‖b⊥‖. (2)
Besides a phononic contribution with material constant cphon and a phasonic contri-
bution with cphas, a coupling term is present with ccoupl. According to experiment we
assume cphon ≫ cphas ≈ ccoupl. Since stable dislocations are those with the lowest energy,
we have to minimise ‖b‖‖. We will discuss this in parallel for dislocations in the Ξ- and
the T -approximants. The minimization yields b2 = b3 in both cases. Furthermore we have
b1 = −τ−1b2 for b3DΞ and b1 = τb2 for b3DT . Here we approximate τ−1 by the fractions
Fm−1/Fm and τ by the fractions Fm+1/Fm respectively. (Fm)m∈N are the Fibonacci num-
bers with start values F1 = F2 = 1. Finally the Burgers vectors of stable dislocations are:
Ξ = (Fm−1,−Fm,−Fm) and b
T = (Fm+1, Fm, Fm). Interestingly they correspond to the
same six-dimensional Burgers vectors:
Ξ = b
T = (0, 0,−Fm−2, Fm−1, Fm−2, Fm−1). (3)
Hence it suffices to consider both cases together for the rest of our calculations. The
phononic component b‖ = b
Ξ = b
is perpendicular to the phason-planes (in the vertical
direction in Fig. 1). We get ‖b‖‖ = τ−mt3D and ‖b⊥‖ = τm−3t3D. Substituting this into
(2), we have E = [cphonτ
−2m+3 + cphasτ
2m−3 + ccoupl] τ
−3(t3D)2. There is a minimum for
cphon/cphas = τ
4m−6. This determines the Burgers vector with lowest energy for given
values for the material constants cphon and cphas. However it has to be noted that these are
not necessarily identical for the Ξ- and the T -approximants.
Tilings of the ξ′3- and the T3-phase with m = 4 dislocations have been calculated
(Fig. 2). They show large rearrangements of the tiles due to the phasonic component b⊥
and negligible deformations of the tiles due to the smaller phononic component b‖. The
dislocations are also dislocations in the metastructure of the phason planes. Therefore
Klein et al. (1999), who discovered these dislocations in HRTEM images of the ξ′2-phase,
named them metadislocations. The fact, that the experimentally most often observed
metadislocations are those withm = 4 suggests cphon/cphas = τ
10 ≃ 123. We do not know of
Burgers vector determinations or observations of metadislocations in the T -approximants,
but dislocations with the Burgers vectors (3) are also the ones most often observed in the
i-phase (Rosenfeld et al. 1995).
4 Discussion and conclusion
In the ξ′n- and the Tn-phases the phasonic degree of freedom is related to the movement
of the phason planes. Since there are no phason planes in the ξ-, ξ′-, R-, and T -phase,
the phasonic degree of freedom cannot be excited locally. However metadislocations can
exist in the ξ′- and the T -phase, but not in the ξ- and the R-phase. It can be shown, that
there is no consistent way to introduce dislocations with phasonic components in the latter
phases.
A motion of the metadislocation (like the motion of any dislocation in a quasicrystal
or large unit cell approximant) is necessarily accompanied by diffusion in the form of tile
rearrangements. The motion is possible by climb in direction of the phason planes or by
glide perpendicular to them. During the climb motion new phason planes are created
(or dissolved) behind the dislocation core. A large number of metadislocations moving
through the ξ′- or T -phase could even lead to a phase transformation to the ξ′n- or Tn-phases
making the phasonic degree of freedom continously excitable. This is affirmed by HRTEM
images of phase boundaries between the ξ′- and the ξ′2-phase formed by metadislcations
(Heggen and Feuerbacher 2005).
On the other side, glide motion does not change the number of phason planes. At least in
the ξ′- and the T -phase glide motion seems unprobable, since the phason planes running out
of the dislocation core would have to be dragged along, while climb motion only needs a re-
construction of the tiling near the dislocation core. (Similar arguments leading to the same
conclusion, as well as newer experimental work are presented in Feuerbacher and Heggen
(2005).) We have to note, that there are no direct observations of metadislocation motion
in approximants yet, although in the i-phase dislocations with identical Burgers vectors
have been shown by in-situ observations to move by climb (Mompiou et al. 2004).
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	Introduction
	Tiling models
	Metadislocations
	Discussion and conclusion
