A unified projection formalism for the Al-Pd-Mn
quasicrystal Ξ-approximants and their metadislocations
M. Engel and H.-R. Trebin
Institut für Theoretische und Angewandte Physik, Universität Stuttgart,
Pfaffenwaldring 57, D-70550 Stuttgart, Germany
email: mengel@itap.physik.uni-stuttgart.de
June 23, 2021
Abstract
The approximants ξ, ξ′ and ξ′n of the quasicrystal Al-Mn-Pd display most in-
teresting plastic properties as for example phason-induced deformation processes
(Klein, H., Audier, M., Boudard, M., de Boissieu, M., Beraha, L., and
Duneau, M., 1996, Phil. Mag. A, 73, 309.) or metadislocations (Klein, H.,
Feuerbacher, M., Schall, P., and Urban, K., 1999, Phys. Rev. Lett., 82,
3468.). Here we demonstrate that the phases and their deformed or defected states
can be described by a simple projection formalism in three-dimensional space - not
as usual in four to six dimensions. With the method we can interpret microstructures
observed with electron microscopy as phasonic phase boundaries. Furthermore we
determine the metadislocations of lowest energy and relate them uniquely to exper-
imentally observed ones. Since moving metadislocations in the ξ′-phase can create
new phason-planes, we suggest a dislocation induced phase transition from ξ′ to ξ′n.
The methods developed in this paper can as well be used for various other complex
metallic alloys.
1 Introduction
A large number of periodic and quasiperiodic phases have been observed in the ternary
Al-Pd-Mn system (Klein, Durand, and Audier 2000). Apart from the stable icosahedral
phase (i-phase) there is a stable decagonal phase (d-phase) with 1.2 nm periodicity and a
metastable d-phase with 1.6 nm periodicity, found by Tsai et al. (1991). The orthorhombic
Ξ-approximants (ξ, ξ′ and ξ′n) of the i-phase form a class of closely related periodic phases
and can be viewed as approximants of this 1.6 nm d-phase, as they have the following
features in common with the d-phase:
(a) They are arrangements of columns of Mackay-type clusters and intermediary atoms
(Sun and Hiraga 1996). These clusters consist of about 52 atoms, placed on concentric
shells of icosahedral symmetry as shown for the ξ′-phase by Boudard et al. (1996).
(b) The columns are in registry, i.e. the compounds also have a layer structure.
The clusters contain about 80% of the atoms. Therefore a coarsened structural de-
scription makes sense where only the projections of the cluster columns along the column
lines are marked. They lead to two-dimensional tilings, which are characteristic for the
respective approximant phase. The vertices of the tiles correspond to the projections of
the cluster columns.
The tilings contain flattend hexagons, which for the ξ-phase are aligned parallel (fig-
ure 1(a)), for the ξ′-phase are staggered in two orientations (figure 1(b)). In the ξ′-phase
isolated combinations of a pentagon and a banana-shaped nonagon (figure 1(c)) are ob-
served, along which the orientation of the hexagons is inverted and which are able to move
by flips. Klein et al. (1996) therefore have termed them phason-lines, following the nota-
tion of related defects in quasicrystals. In bending experiments these phason-lines order
into periodically aligned phason-planes with 1, 2, . . . , n rows of hexagons in between (fig-
ure 1(d) and (e)), forming periodic superstructures. Here we propose to name these phases
ξ′2-, ξ
3-, . . ., ξ
n+1-phases, the reason for the index-shift being given later. Most observed is
the ξ′2-phase, which is also known as Ψ-phase (Klein 1997).
The ξ′n-phases can be considered striped defect lattices. The stripes, which are the
phason-planes, can bend and move, varying their distances (Beraha et al. 1997). Klein
et al. (1999) have observed dislocations in the stripe pattern and have called them metadis-
locations. They are the characteristic textures of partial dislocations in the basic tiling.
In this article we are developing a simplified projection formalism to describe all the
Ξ-phases. It is derived from the hyperspace method of the quasiperiodic phases. As a
consequence, phasonic degrees of freedom can exist in the Ξ-phases which allow move-
ments of the phason-planes. We determine those hyperlattice Burgers vectors which label
the partial dislocations of lowest energies and relate them to the Burgers vectors of the
metadislocations. Indeed, these are the only ones which are observed. The charming fea-
ture of our formalism is that the simplest model is working in three space so that all steps
are easily imaginable.
2 Geometrical models for i-Al-Pd-Mn and its approx-
imants
The atomic positions for a quasicrystal can be described by decorating the lattice points of
a periodic hypercrystal with atomic surfaces and marking the points where these are inter-
sected by a planar cut space (cut-method). An equivalent method is the strip-method (Katz
and Duneau 1986). Approximants are constructed by using inclined cut spaces (Duneau
and Audier 1994). For a detailed explanation see also Gratias, Katz, and Quiquandon
(1995).
0.78 nm
Figure 1: Tilings of four different approximant phases of the 1.6 nm d-phase. (a) Parallel
alignment of hexagons in the ξ-phase. (b) Staggered arrangement of hexagons in the ξ′-
phase. Two hexagon rows are marked grey. (c) The hexagons H occur in two different
orientations. The other tiles, the pentagon P and the nonagon N can only be observed in
combination, called phason-line. The edge length of the tiles is: t6D = 0.78 nm. (d) In the
ξ2-phase there is one row of hexagons between two phason-lines, also called phason-planes.
(e) In the ξ3-phase there are two rows of hexagons between neighbouring phason-planes.
The arrangement of hexagons is flipped at a phason-line as highlighted in the figure.
2.1 Sixdimensional cut-method for the i-phase and approximants
A six-dimensional hyperspace with orthogonal basis vectors e6D1 , . . ., e
6 of length l
0.645 nm was used by Katz and Gratias (1994) to model the i-phase of Al-Pd-Mn. The
hyperspace was decorated with three different atomic surfaces positioned on different nodes
in a face-centred lattice, namely
• even nodes: n0 = {(z1, . . . , z6), zi ∈ Z |
i zi = 0 mod 2},
• odd nodes: n1 = {(z1, . . . , z6), zi ∈ Z |
i zi = 1 mod 2},
• even body-centre nodes: bc0 = {(z1 + 12 , . . . , z6 +
), zi ∈ Z |
i zi = 0 mod 2}.
The cut space is spanned by three vectors, whose components in the above basis are
(τ = 1
5 + 1) is the golden mean):
a6Di = (1, τ, 0,−1, τ, 0), b6Di = (τ, 0, 1, τ, 0,−1), c6Di = (0, 1, τ, 0,−1, τ). (1)
Beraha et al. (1997) applied the cut-method for the three-dimensional atomistic descrip-
tion of the phases ξ and ξ′ as approximants of the icosahedral phase. In order to position
each single atom correctly, they had to modify the atomic surfaces slightly. It turned out
that the centre volumes of the n0 atomic surfaces, which lie again on a six-dimensional
face-centred lattice, correspond to the cluster centre atoms. This way the model is reduced
to three-dimensional tilings for the cluster positions only.
For the ξ′-phase Beraha et al. have derived the vectors that span a unit cell of the
inclined cut space. We add the vectors for the inclined cut spaces of the ξ-phase and the
ξ′n-phases:
a6Dξ = (0, 1, 1,−1, 0, 1), b6Dξ =
(5, 1, 1, 1, 1,−1), c6Dξ = (0, 0, 1, 1,−1, 1) (2a)
a6Dξ′ = (0, 2, 1,−2, 1, 2), b6Dξ′ = b6Dξ , c6Dξ′ = (0, 0, 1, 1,−1, 1) (2b)
a6Dξ′n = (0, 2, 1,−2, 1, 2), b
= b6Dξ , c
= (0, 0, 2n+ 1, 2n,−2n− 1, 2n) (2c)
The b6Dξ -vector is the same for all the phases, because it marks the periodicity in the
tenfold direction of the d-phase, which coincides with the column line of the Mackay-type
clusters. By projecting in direction of b6Dξ , two-dimensional tilings like those in figure 1
can be obtained. The edge length of the tiles is t6D = 1
τ + 2 l6D = 0.78 nm. It can
be calculated by projecting all connection vectors e6Di + e
j of neighbouring n0-sites onto
the tiling plane. The shortest projections have length t6D.
Since the six vectors a6Dξ , c
ξ , a
ξ′ , c
ξ′ , a
and c6Dξ′n = 2nc
ξ′ − a6Dξ′ + 2a6Dξ lie in a
three-dimensional subspace of the six-dimensional hyperspace, a description of the tilings in
a three-dimensional hyperspace is possible by the cut-method. The model can be simplified
by using a simple cubic lattice in three-space, as will be shown in detail in section 2.4. We
can arrive at this conclusion directly after substituting the tiling of hexagons, pentagons
and nonagons by a tiling of rhombs.
2.2 Rhombic substitution tiling
The new tiles are the thin and thick Penrose rhombs which in general are spanned by the
vectors of a regular five-star. The interior angles of the tiles are multiples of 36◦.
As shown in figure 2, a hexagon of the original tiling is substituted by a thick rhomb,
and a nonagon/pentagon combination is substituted by a combination of a thin and a thick
rhomb.1 Therefore we will refer to the thin rhombs as phason-lines. A row of hexagons in
staggered orientation is substituted by a row of alternating thick rhombs, and a phason-
plane is substituted by a combination of a row of alternating thick rhombs and a row of
thin rhombs, which again will be called phason-plane. Hence the number of rows of thick
rhombs between two phason-planes in the new tiling for the ξ′n-phase is exactly n.
Thus the tilings for the column positions of the Ξ-phases emerge as approximants of
the Penrose tiling. However, only rhombs of three orientations show up. To span these
only three prongs of the five-star are required. This is another argument why we can apply
a projection formalism in a three-dimensional hyperspace which was mentioned above and
is elaborated further below.
To model the lattices of all Ξ-phases, their phasonic degrees of freedom and their
metadislocations we found it suitable to resort to a modified cut and projection method.
To our knowledge it has not been applied yet in the literature. It makes use of atomic
hypervolumes and requires a short section.
2.3 Atomic hypervolumes for a geometrical description in hyper-
space
The method will be explained by the example of the well known one-dimensional Fibonacci-
chain: The two-dimensional hyperspace is partitioned into equal unit cells, called atomic
volumes (in general atomic hypervolumes) as shown in figure 3(a).2 For the construction
of an approximant, two different lines are needed: the cut line E (in general cut plane or
cut space) and the projection line E (in general projection plane or projection space, also
named physical space). Those cells that are cut by E are selected (marked grey). The
middle point of those cells is projected onto the projection line.3 The projection leads to
two different intervals on E, either a small one (S), when the neighbouring selected cells
meet vertically, or a large one (L), when they meet horizontally, forming the tiles of the
Fibonacci-chain.
1A similar tiling has been presented by Klein et al. (1996). They used thick rhombs to model the ξ- and
ξ′-phase and small hexagons for the phason-lines. However no description in hyperspace was presented,
explaining the arrangement of the tiles.
2As a generalisation several overlapping atomic volumes could be used, as long as the number of times
a point is covered is constant for the hyperspace. This way the closeness condition (Frenckel, Henley, and
Siggia 1986) of the original cut-method is automatically fulfilled. But we will only make use of the method
of atomic volumes for the canonical case of a hyper-cubic lattice Zn and the unit cells as atomic volumes.
3Any other choice is possible too, as long as it is the same point for each cell, since a different selection
only leads to a global translation of the tiling.
1.26 nm
(c) (d)
(b)(a)
Figure 2: New tilings can be created by substituting the original tiles with thin Penrose
rhombs RN, called phason-lines, and thick Penrose rhombs RB. The ξ-phase (a) and the
ξ′-phase (b) are built only with the thick rhombs, while for the ξ′2-phase (c) and the ξ
phase (d) also the thin rhombs are needed. Between two phason-planes in the ξ′n-phase
there are n rows of alternating thick rhombs (marked grey). The new tiles are shown in (e).
The edge length of the tiles is: t5D = 1.26 nm.
The orientation of E determines the shape of the tiles. In the example of the Fibonacci-
chain, it determines the ratio of S to L. In contrast, the orientation of E fixes the arrange-
ment of the tiles. Since different approximants of the same quasiperiodic phase are built
from the same tiles, but with different arrangement of the tiles, the only difference in the
construction is the varying orientation of E . A rational slope of the cut line leads to a
periodic tiling, while an irrational slope generates a quasiperiodic tiling. The special case
of E and E being identical is the original cut-method as mentioned above.
A geometrical restriction on the orientation of E is imposed by the fact, that projected
tiles must not overlap. In the example of the Fibonacci-chain this means, that the slope
of E must be positive. For a counterexample when it is not fulfilled see figure 3 (b). With
this method we now construct the tilings of the cluster projections.
In the original cut-method the shape of the atomic surfaces depends on the orientation
of the cut space. For the Fibonacci-chain the atomic surfaces are the projections of the
unit cells on the orthogonal complement of E . Because we will deal with cut spaces with
spatially varying orientation in this paper, we would need a varying shape of the atomic
surfaces as well. The method of atomic hypervolumes avoids these problems, as the atomic
hypervolumes themselves do not vary.
2.4 Three- and five-dimensional hyperspace for approximants of
Al-Pd-Mn
Because the rhombic tilings for the Ξ-phases are approximants of the Penrose tiling they
can be described in a five-dimensional hyperspace with basis {e5D1 , . . . , e5D5 }, the hyper-
cubic lattice Z5 and the unit cells as atomic hypervolumes. The projection plane of the
Penrose tiling is spanned by the vectors
a5Dp =
sin(2π 0
), sin(2π 1
), sin(2π 2
), sin(2π 3
), sin(2π 4
c5Dp =
cos(2π 0
), cos(2π 1
), cos(2π 2
), cos(2π 3
), cos(2π 4
and the projection matrix is obtained by writing these vectors normalised in the rows:
π5D‖ =
sin(2π 0
) sin(2π 1
) sin(2π 2
) sin(2π 3
) sin(2π 4
cos(2π 0
) cos(2π 1
) cos(2π 2
) cos(2π 3
) cos(2π 4
. (4)
The projections f i = π
i of the five-dimensional basis vectors give a regular five-star
as drawn in figure 4(a). The rhomb tiles are the projection of two-dimensional faces of the
five-dimensional unit cells. Each is spanned by two vectors of the five-star.
The edge length of the rhomb tiles can be calculated from the edge length of the
pentagon/hexagon/nonagon-tiles as shown in figure 2(e): t6D = 2 cos(72◦) t5D = τ t5D.
This gives: t5D = 1
τ + 2l6D = 1.26 nm, leading to the length of the basis vectors
in the five-dimensional hyperspace: l5D = τ
τ + 2 l6D = 1.99 nm.
A basis {a5D, c5D} for the cut plane of a periodic approximant can be determined with
a basis of the unit cell of the tiling. This is done in figures 4(b)-(e) for various approximant
Figure 3: (a) Construction of an approximant of the Fibonacci-chain by the method of
atomic volumes. A cut line E and a projection line E are required (see text). (b) If the
slope of E is negative, the projected tiles overlap. This case is geometrically not allowed.
In the bottom left and top right corner a small part of the quasiperiodic Fibonacci-chain
is drawn.
(e)(a) (d)
f 3f 4
f5 − f 2
4f 1 + f 2 + f 5
f5 − f 2
6f 1 + f 2 + f 5
f5 − f 2
Figure 4: (a) The projected basis vectors f i lie on a regular five-star. The three rhombs of
the tilings in figure 2 can be constructed with f 1, f 2 and f 5. Unit cells of approximants
and their bases are shown: the ξ-phase (b), the ξ′-phase (c), the ξ′2-phase (d) and the
ξ′3-phase (e).
phases:
a5Dξ = (0, 0, 0, 0, 1), c
ξ = (1, 0, 0, 0, 0), (5a)
a5Dξ′ = (0,−1, 0, 0, 1), c5Dξ′ = (1, 0, 0, 0, 0), (5b)
a5Dξ′n = (0,−1, 0, 0, 1), c
= (2n, 1, 0, 0, 1). (5c)
We now want to derive the transition matrix T 6D5D from the five- to the six-dimensional
hyperspace which maps the basis vectors a5D and c5D on a6D and c6D. The first column
of T 6D5D is fixed by the relation c
ξ = T
ξ . The other columns follow from symmetry
considerations. The fivefold rotation of the hyperspaces that fixes the projection plane is
for the five- and six-dimensional model:
R5D =

0 1 0 0 0
0 0 1 0 0
0 0 0 1 0
0 0 0 0 1
1 0 0 0 0
 , R6D =

1 0 0 0 0 0
0 0 1 0 0 0
0 0 0 1 0 0
0 0 0 0 0 −1
0 1 0 0 0 0
0 0 0 0 −1 0
 . (6)
Observe, that R6D does not change the vector b6Dξ . The symmetry operation in the two
hyperspace must match: R6DT 6D5D = T
5D. This determines the (i + 1)-th column of
T 6D5D via
c6Dξ = [R
6D]−i ◦ T 6D5D ◦ [R5D]i c5Dξ (7)
resulting in:
T 6D5D =

0 0 0 0 0
0 −1 −1 1 1
1 0 −1 −1 1
1 1 0 −1 −1
−1 −1 1 1 0
1 −1 −1 0 1
 . (8)
After fixing a coordinate system for the atomic hypervolumes in the five-dimensional
hyperspace, the approximant phases can appear in five different orientations corresponding
to a cyclic permutation of the basis vectors. Each orientation of the phases ξ′ and ξ′n is
characterised by the orientation of the thin rhombs. By fixing their orientation, a lower-
dimensional hyperspace can be used. Since the third and fourth components of the basis
vectors in (5) vanish, a description in a three-dimensional hyperspace with cubic lattice Z3
(lattice parameter l5D = l3D) and the unit cell as atomic volume is possible by omitting
these components. The transition matrix is simply:
T 5D3D =

1 0 0
0 1 0
0 0 0
0 0 0
0 0 1
 . (9)
For the Ξ-phases this three-dimensional model leads to the same tilings as the five-dimensional
model, but only one orientation of the thin rhombs and two orientations of the thick rhombs
can show up. The projection matrix π3D‖ is derived from π
‖ again by omitting the third
and fourth components:4
π3D‖ =
sin(2π 0
) sin(2π 1
) sin(2π 4
cos(2π 0
) cos(2π 1
) cos(2π 4
2 + τ −1
2 + τ
τ−1 1
. (10)
Bases for the cut planes in the three-dimensional model are given by
a3Dξ = (0, 0, 1), c
ξ = (1, 0, 0), (11a)
a3Dξ′ = (0,−1, 1), c3Dξ′ = (1, 0, 0), (11b)
a3Dξ′n = (0,−1, 1), c
= (2n, 1, 1). (11c)
2.5 Relationships of the hyperspaces
So far the three- and five-dimensional hyperspace have been introduced only phenomeno-
logically for the description of tilings of the approximant phases. Now we want to show
4This projection is not orthogonal, while the projection π5D‖ is orthogonal. That is a consequence of
the omission of two hyperspace dimensions.
how they are related to the original six-dimensional hyperspace. This is meant as a clari-
fication of hyperspace geometry. Mathematically, the formation of the approximants from
the icosahedral quasicrystal proceeds in two steps:
(i) The configuration in direction of a fivefold rotation axis b is changed, while the
configuration perpendicular to this direction in the plane spanned by a and c is
unaltered, leading to a pentagonal or decagonal quasicrystal. This step cannot be
described simply by a reorientation of the cutplane. Additionally a relaxation of the
atoms in direction of b is necessary as shown by Beraha et al. (1997).
(ii) The configuration in the plane spanned by a and c is changed, while the configuration
in direction of b is unaltered. This step is accurately described by a reorientation of
the cutplane as used in this paper.
In step (i) a fivefold symmetry of the icosahedral quasicrystal is conserved. The fivefold
rotation R6D (equation (6)) operates trivially on a two-dimensional subspace U1 spanned
by (1, 0, 0, 0, 0, 0), (0, 1, 1, 1, 1,−1) and as a true rotation on the four-dimensional orthogo-
nal complement U2 spanned by (0, 0, 1, 1,−1, 1), (0,−1, 0, 1,−1,−1), (0,−1,−1, 0, 1,−1),
(0, 1,−1,−1, 1, 0). So for quasicrystals and approximants the lattice vector b is projected
from U1, and the vectors a and c are projected from U2.
The restriction of Z6 onto U2 leads to a non-cubic four-dimensional lattice (the A4
root lattice) that could in principle be used for the construction of the Penrose tiling
and its approximant tilings. By adding a one-dimensional complementary space ∆, these
tilings can be constructed from the Z5-lattice. This is well-known and the usual way to
describe the Penrose lattice. The relation of the five-dimensional hyperspace to the three-
dimensional hyperspace is described in section 2.4. Figure 5 summarizes all the hyperspace
relationships.
3 Phasonic degrees of freedom
Besides standard elastic (phononic) degrees of freedom, quasicrystals have additional de-
grees of freedom (Socolar, Lubensky, and Steinhardt 1986), which originate from the fact
that the cut space is embedded in hyperspace. Local excitations of these so-called pha-
sonic degrees of freedom, correspond to continuous displacements of the cut space along
the direction orthogonal to it in hyperspace. The direction of the excitation is understood
as the direction of the displacement.
The number of the phasonic degrees of freedom for a n-dimensional hyperspace and a
d-dimensional cut space is equal to n− d. As will be shown, phasonic degrees of freedom
are also possible for approximant phases, although then they are not any more continuous
degrees of freedom. For convenience we nonetheless continue using this notation.
U1 ⊕ U2 (6D)
U1 (2D) U2 (4D) U2 ⊕∆ (5D) 3D
Span(b) (1D) Span(a, c) (2D)
Span(a, b, c) (3D)
T 6D5D
T 5D3D
hyperspaces
physical spaces
Figure 5: Relationship of the hyperspaces and the physical spaces. The five-dimensional
space is built on top of the four-dimensional space U2 perpendicular to the fivefold two-
dimensional plane U1. It is not a (full) subspace of the original six-dimensional space. Solid
arrows indicate projections (subspace relationship). Dashed arrows indicate inclusions.
T 6D5D |U2 is an inclusion and T 6D5D |∆ = 0 the zero mapping.
Figure 6: Local excitation of the phasonic degree of freedom for an approximant of the
Fibonacci-chain. The excitation rearranges the sequence of the tiles.
3.1 Restriction to excitable degrees of freedom
In the example of the Fibonacci-chain, a local excitation of the phasonic degree of freedom
is shown in figure 6. Because of the displacement the cut line E now selects different atomic
volumes. In the tiling this leads to a local rearrangement of the tiles, called phasonic jumps.
In general the excitations are considered small and distributed over a wide tiling region,
so that the orientational deviation from the planar cut space is small.
A special case has to be examined separately: If the cut plane of an approximant is
parallel to boundaries of atomic volumes, it is possible that a local excitation of phasonic
degrees of freedom leads to overlapping tiles. If, for example in the Fibonacci-chain, the cut
line of the approximant has slope 0, a local excitation always creates regions of the cut line
having positive as well as negative slope. The regions of negative slope lead to overlapping
tiles, as shown in figure 3(b) earlier. In such a case, accordingly, local displacements of the
cut plane are geometrically not allowed and the phasonic degree of freedom is termed not
excitable.
With regard to the approximants of the 1.6 nm d-(Al-Pd-Mn) in the five-dimensional
model, the phasonic degrees of freedom of the Ξ-phases are not excitable in the directions
of e5D3 and e
4 . So for a full description of these phases the three-dimensional hyperspace
with basis {e3D1 = e5D1 , e3D2 = e5D2 , e3D3 = e5D5 } and only one phasonic degree of freedom is
sufficient.
This last phasonic degree of freedom is excitable in the ξ′n-phases, leading to new
phenomena. But it is not excitable in the ξ- and the ξ′-phase. (The cut plane of the ξ′-
phase, for example, is parallel to the e3D1 -axis). So the phases ξ and ξ
′ behave like normal
periodic crystals, having only phononic degrees of freedom.
It is a pleasant aspect of the three-dimensional hyperspace that the construction for-
malism can easily be visualised. Thus in the next sections we can present the cut plane
n1 × n2 n2 (b)
n1 × n2
Figure 7: (a) Two different cut planes with normal vectors n1 and n2. They have the line
g with direction vector n1 × n2 in common. (b) A transition between the two phases is
realised by adjusting the orientation of the cut plane continuously (see text).
for phase boundaries and dislocations as two-dimensional curved surfaces.
3.2 Phasonic phase boundaries
Different approximant phases are characterised by different orientations of the cut space.
As a feature of the phasonic degrees of freedom, a spatially dependent continuous transfor-
mation from one orientation of the cut space to another is possible, leading to a phasonic
phase boundary.
Let two phases be characterised by the normal vectors n1 and n2 of their cut planes
E1 and E2 (figure 7(a)). For calculating the cut plane E of the phasonic phase boundary
we choose the coordinate system in a way, that the line g = E1 ∩E2 with direction n1×n2
runs through the origin. In the limit far away from g on the side of phase 1 the new cut
plane E contains the vector v1 = n1 × (n1 × n2), while on the other side E the vector
v2 = n2 × (n1 × n2). By using a transition function f : R→ [0, 1] with properties
f(x) = 0, lim
f(x) = 1 and f(0) = 1/2 (12)
the cut plane for the phase boundary E as shown in figure 7(b) is parametrised by:
n1 × n2
‖n1 × n2‖
f(t)v2 + (1− f(t))v1
‖f(t)v2 + (1− f(t))v1‖
; s, t ∈ R
. (13)
Klein et al. (1996) have taken a transmission electron micrograph of a phasonic phase
boundary (figure 8(a)), without interpreting it as such. The phason-lines can be seen as
dark contrasts. Phase 1 on the left side is one of the phases ξ(α), 0 ≤ α ≤ 1 with α ≈ 0.8,
by which we denote an intermediate phase of a ξ- and a ξ′-approximant. Its cut plane
is spanned by aξ(α) = (0,−α, 1) and cξ(α) = (1, 0, 0). Special cases are: ξ(0) = ξ and
ξ(1) = ξ′. Phase 2 on the right side is the ξ′2-approximant.
The tiling for this phasonic phase boundary from ξ(0.8) to ξ′2 is calculated as explained
above and shown in figure 8(b). For the transition function we used:
f(x) =
0 , x ≤ 0
arctan(x)
, x > 0.
In the region of the ξ(0.8)-phase (x < 0) the cut plane is flat, since in the ξ(α)-phases the
phasonic degree of freedom is not excitable, just as in the ξ- or the ξ′-phase. In the region
of the ξ′2-phase (x > 0) excitations of the phasonic degree of freedom show as bendings of
the phason-planes.
As a direct consequence of the hyperspace description the orientation of the phasonic
phase boundary is completely determined by the involved approximant phases. On the
other side, by observing this orientation we can identify the approximant phases, labelled
by the parameter α.
4 Dislocations: Energetic consideration
In quasicrystals and approximants dislocations do exist and are characterised by a unique
Burgers vector b, which now is a vector of the hyperlattice. A dislocation is accompanied
by a phonon- and a phason-strain field, due to the phononic part b‖ = π‖b and the
phasonic part b⊥ = π⊥b (π⊥ projects on the orthogonal complement of E), respectively,
of the Burgers vector. If the phasonic part vanishes, the dislocation is a classical one
and can be described without the hyperspace methods. Since the lattice constants of
periodic approximants are large, classical dislocations have gigantic Burgers vectors with
huge phonon-strains. They are energetically unfavourable and will not be observed.
By extending the linear theory of elasticity to hyperspace (approximating the phasonic
degree as continuous), the line energy E of a dislocation grows quadratically with increasing
lengths ‖b‖‖ and ‖b⊥‖.5 Assuming isotropy in the phononic and phasonic part, E can be
expressed as:
E = cphon‖b‖‖2 + cphas‖b⊥‖2 + ccoupl‖b‖‖‖b⊥‖. (15)
Besides a phononic contribution with elastic constant cphon and a phasonic contribution
with cphas, a coupling term is present with ccoupl.
For estimating the different contributions in (15) the elastic constants in the approx-
imant phases are assumed to be comparable to those of i-Al-Pd-Mn. For the icosahedral
phase the phononic elastic constants have been determined by Amazit et al. (1995) from
sound-propagation. The phasonic elastic constants have been measured with neutron and
x-ray scattering experiments by Letoublon et al. (2001). From the experimental values it
5There is an ongoing discussion whether the elastic energy grows linear (locked state) or quadratically
(unlocked state) in the phasonic strain, see the review by Edagawa (2001). However at higher temperatures,
when dislocations can form and move, the unlocked state is entropically stabilised.
6In several papers the coupling term is neglected, although its appearance is a consequence of the linear
theory of elasticity.
Figure 8: Phasonic phase boundary between the phase ξ(0.8) on the left side and the phase
ξ′2 on the right side. (a) Transmission electron micrograph. The darker spots correspond
to phason-lines (Klein et al. 1996). (b) Calculated tiling for the three-dimensional model
(see text).
can be concluded, that cphas is a few orders of magnitudes smaller than cphon. Furthermore
computer simulations (Koschella et al. 2002) suggest that ccoupl has about the same value
as cphas. So the largest contribution to E comes from the phononic part. In the tiling
picture this means that it is energetically more favourable to rearrange parts of the tiling
than to deform tiles.
5 Dislocations in the ξ′n-phases
The construction method for a dislocation in hyperspace has been discussed e.g. by Bohsung
and Trebin (1989). A dislocation with Burgers vector b forces upon the cut plane a dis-
placement field u with a phononic and a phasonic part. In a first approximation the
displacement field can be distributed isotropically in physical space and becomes in spher-
ical coordinates:
u(r, ϕ) =
b. (16)
5.1 Metadislocations in the three-dimensional model
The steps involved in the construction of a dislocation are visualised in figure 9. The
distorted cut plane has a strain jump of b3D along a radial line as shown in figure 9(a).
The middle points of the atomic volumes selected by the cut plane form a staircase surface
as shown in figure 9(b). This surface is projected with π3D‖ onto the projection plane
(figure 9(c)). Up to now only the phason-strain of the dislocation has been taken regard of
and results in a rearrangement of the tiles. The gap in the tiling is caused by the jump line
in the cut plane. In a last step the phonon-strain is introduced (figures 9(d), (e)), closing
the gap and deforming the tiles. In (figure 9(f)) a Burgers circuit is constructed. The sides
of the tiles correspond to projected basis vectors f i of the hyperlattice. In the example
the lines A, C and B, D cancel each other, only E is left and is lifted to the Burgers vector
b3D = (−2, 1,−3) = −2e3D1 + e3D2 − 3e3D3 .
Consider now an arbitrary dislocation with b3D = (b1, b2, b3), bi ∈ Z. Its phononic
component
b‖ = π
τ + 2(b2 − b3)
τ−1(b2 + b3)
. (17)
must be small for low dislocation energy, as shown in the last section. This is achieved,
if b2 = b3 and
≈ −τ . The best approximating rational values for τ are given by frac-
tions Fm
of successive Fibonacci numbers (Fm)m∈N = (1, 1, 2, 3, 5, 8, 13, 21, . . .) defined
by Fm+1 = Fm−1 + Fm with start values F1 = F2 = 1. They have the following properties,
which can be proved by induction:
(i) Fmτ + Fm−1 = τ
m (18a)
(ii) Fm+1 − Fmτ = (−τ)−m (18b)
(iii) Fm =
5(τm − (−τ)−m) (18c)
(b) (c)
(d) (e)
3f2 3f3
b3D = (−2, 1,−3)
Figure 9: Construction of a dislocation in the three-dimensional model. (a) The displaced
cut plane E of the ξ′3-phase around the dislocation core. (b) Staircase surface of those atomic
volumes, that are cut by E . (c) Projection of the staircase surface onto the projection plane
E. (d) The new tiling around the dislocation core with rearrangements of the tiles due to
phason-strain. (e) The gap in the tiling is closed by phonon-strain. (f) The Burgers vector
b3d can be obtained by performing a Burgers circuit.
Therefore the best candidates for Burgers vectors of low energy dislocations are
b3D =
 Fm−1−Fm
 with b‖ = 1
Fm−1 − τ−1Fm
and length:
‖b‖‖ = τ−m
10 l3D = τ−m · 1.26 nm. (20)
With (8) and (9) we get the 6D Burgers vector:
b6D = T 6D5D T
3D = (0, 0,−Fm−2, Fm−1, Fm−2, Fm−1). (21)
The phasonic component for the six-dimensional hyperspace can be calculated with (18):
‖b⊥‖ =
‖b6D‖2 − ‖b‖‖2 =
2 [(Fm−1)2 + (Fm−2)2] (l6D)2 −
τ−2m(l3D)2
= τm−3
10 l3D = τm−3 · 1.26 nm, (22)
yielding the strain accommodation parameter ζ, i.e. the ratio of the phasonic and the
phononic component of the Burgers vector (Feuerbacher et al. 1997):
= τ 2m−3. (23)
Tilings for dislocations with m = 4 and m = 5 are shown in figures 10(a) and (c). The
dislocation cores appear as triangles which can be used for Burgers circuits. Phason-planes
come in from the left side of the tiling, ending at the flat side of the triangle, their number
being equal to that of the tiles there, which is 2Fm. Whereas the phononic part of the
Burgers vector is much smaller than the side length l3D of the rhombs, the phasonic part
is so large that it leads to a bending of the phason-planes over a huge area of the tiling.
The arrangement of the phason-lines can be viewed as a metastructure which, in the
simple case of a dislocation free ξ′n-phase is a striped centred rectangular lattice. A par-
tial dislocation in the tiling leads to a dislocation in the striped metastructure, named
metadislocation. A dislocation with a three-dimensional Burgers vector as in (19) is called
metadislocation of type m. In contrary to the dislocation constructed in hyperspace, a
metadislocation is an ordinary dislocation in the metastructure with a two-dimensional
Burgers vector.
The Burgers vector bmeta of a type m metadislocation is oriented in the direction of
the lattice vector cξ′n = π
. Since a lattice cell contains two phason-planes, a type m
metadislocation has Fm inserted rows of lattice cells in this direction, leading to a Burgers
vector
bmeta = Fmπ
2n+ τ−1
of length
‖bmeta‖ = l3DFm(2n+ τ−1) = Fm(2n+ τ−1) · 1.26 nm. (25)
The most frequent dislocations are the type 4 metadislocations in the ξ′2-phase. They
have been observed by electron microscopy (Klein et al. 1999), see figure 10(b). From
our theory we derive the lengths ‖b‖‖ = 0.184 nm and ‖b⊥‖ = 2.04 nm (ζ = 11.1)
for the dislocation and ‖bmeta‖ = 17.5 nm for the metadislocation, which is larger by
two orders of magnitude. For comparison we present the metadislocation also in the
hexagon/pentagon/nonagon-tiling (figure 10(d)). Transmission electron micrographs of
type 3, 5 and 6 metadislocations have been published, too, by Klein and Feuerbacher
(2003).
5.2 Metadislocations in the five-dimensional model
Other kinds of metadislocations reported by Klein and Feuerbacher (2003) also show a
large scale arrangement of phason-lines, but require the five-dimensional hyperspace for
description, since phason-lines occur in more than one orientation. Here we will discuss
two different types of metadislocations:
1. Metadislocations with Burgers vector of the form: b5D = (Fm,−Fm−1, Fm, 0, 0): The
phononic component is:
b‖ = π
10(−τ)−m
cos(72◦)
sin(72◦)
. (26)
with length ‖b‖‖ = τ−m · 1.26 nm. The Burgers vector is rotated by 72◦ compared
to the metadislocation in (19). A tiling of such a metadislocation with m = 3 is
shown in figure 11(a). Phason-lines now appear in two different orientations. Fm
rows of tilted phason-lines ending at the dislocation core cannot be explained within
the three-dimensional model.
2. Metadislocations with Burgers vector of the form: b5D = (0, Fm−1, Fm−1, 0, Fm): The
phononic component is:
10(−τ)−m−1
cos(−72◦)
sin(−72◦)
. (27)
with length ‖b‖‖ = τ−m−1 · 1.26 nm. A tiling with m = 5 is shown in figure 11(b).
This time Fm−1 rows of tilted phason-lines end at the dislocation core.
For transmission electron micrographs of the two discussed metadislocations see fig-
ure 11(c) and (d). In the first figure the gap extending to the left in the tiling has been
closed by the motion of the phason-planes. In the second figure the metadislocation seems
to have moved up, while building a sack-shaped area of ξ′-phase below. On the sides of
this area, five phason-planes are cut into halves by a region of ξ′-phase.
(c) (d)
Figure 10: (a) Tiling of a type 4 metadislocation in the ξ′4-phase. (b) Transmission electron
micrograph of a type 4 metadislocation in the ξ′2-phase, courtesy by H. Klein and M.
Feuerbacher. (c) Tiling of a type 5 metadislocation in the ξ′4-phase. (d) For comparison
the calculated hexagon/pentagon/nonagon tiling of a type 4 metadislocation in the ξ′4-
phase is shown.
20b3D‖
20b3D‖
Figure 11: Tilings of metadislocations in the ξ′1-phase with Burgers vectors (a) b
(2,−1, 2, 0, 0) and (b) b5D = (0, 3, 3, 0, 5). The phason-strain field is not distributed
isotropic. The phononic component of the Burgers vector is drawn magnified 20 times,
since it is too small otherwise. (c) and (d) show transmission electron micrographs of these
two metadislocations, courtesy by H. Klein and M. Feuerbacher.
(a) (b)
Figure 12: (a) A tiling of a metadislocation in the ξ′-phase with six inserted phason-planes.
In the dislocation tail a strip of ξ′3-phase is created. (b) Cut plane for the metadislocation
in the ξ′-phase. The phason-strain field is not relaxed isotropic, but concentrated in the
dislocation tail, where the phason-lines are observed.
6 Metadislocations in the ξ′-phase
Although phasonic degrees of freedom are not excitable continuously in the ξ′-phase,
metadislocations with phasonic components, which cause a noncontinuous local phason-
strain do exist and have been observed experimentally by Klein and Feuerbacher (2003).
They can be described in the three-dimensional hyperspace. To minimise the dislocation
energy, their Burgers vectors b3D are the same as in the ξ′n-phases (19). For a tiling see
figure 12(a) in the case m = 4. Similar to metadislocations in the ξ′n-phases, a tail of six in-
serted phason-planes ends at the flat sides of the dislocation core. Towards the dislocation
core the phason-planes converge, while far away they run parallel. Contrary to metadislo-
cations in the ξ′n-phases the phason-strain now is not distributed isotropically around the
dislocation core, but concentrated in the region, where phason-lines are observed.
The shape of the cut plane for the metadislocation in the three-dimensional model is
shown in figure 12(b), where the coordinate system is rotated, so that the cut plane Eξ′ of
the ξ′-phase lies horizontally. Because of its small phononic component, the Burgers vector
is almost perpendicular to Eξ′ .
We place the dislocation in the origin and the strain jump of b3D along the y-axis, along
a line in the middle of the strip of phason lines. Inside the dislocation tail and far away
from the core the strain field is
u3D(x, y) =
|x| −
b3D. (28)
The parameter λ controls the width of the dislocation tail. To simulate a convergence of
the dislocation tail λ = λ(y) must be designed decreasing with distance from the dislocation
core. Outside of the dislocation tail, the cut plane is parallel to Eξ′ . The region with the
phason lines can be viewed as a very thin strip of newly created ξ′n-phase, embedded in
surrounding ξ′-phase. We suggest, that these dislocations move in the direction of their
tail, because for the movement perpendicular to it the entire tail would have to be dragged
along. The movement is pure climb, because it is perpendicular to the burgers vector and
the dislocation line. There are indications (Mompiou, Caillard, and Feuerbacher 2004),
that in the i-Al-Pd-Mn quasicrystal dislocations can move by pure climb. Furthermore
the most often observed dislocations in i-Al-Pd-Mn have the same burgers vectors as the
metadislocations in the Ξ-phases (Rosenfeld et al. 1995).
Finally it should be mentioned that Feuerbacher and Caillard (2004) also observed
other dislocations in the ξ′-phase with Burgers vectors that do not lie perpendicular to
the vector b6Dξ′ . These cannot be pictured as two-dimensional tilings, but require a true
three-dimensional description in physical space.
7 Conclusion
In this paper we have introduced a simple projection formalism in five-dimensional hyper-
space for the description of the i-(Al-Pd-Mn) approximants ξ, ξ′ and ξ′n as two-dimensional
tilings. The tilings correspond to the projections of clusters columns. In most cases the
formalism can even be restricted to a three-dimensional hyperspace. The tilings are gen-
erated by cuts and projections through a hyperspace which is partitioned into atomic
hypervolumes.
We have shown that phasonic degrees of freedom in form of continuous displacements
of the cut space do exist in these phases and can be either excitable or not. They play a
fundamental role for phasonic phase boundaries as well as dislocations. In the case of the
ξ- and ξ′-phase no phasonic degrees of freedom are excitable, while in the ξ′n-phases there
is exactly one excitable phasonic degree of freedom, which is connected to the bending of
the phason-planes.
Nevertheless metadislocations in the ξ′-phase can exist, creating phason-planes and
forming a thin strip of ξ′n-phase in their tails while moving by climb. In this way, a consec-
utive motion of metadislocations through the ξ′-phase induces a phase transformation to
a ξ′n-phase, making the phasonic degree of freedom excitable. The same metadislocations
continue to exist in the ξ′n-phases.
It is possible that phasonic degrees of freedom and dislocations with phasonic com-
ponents also occur in various other complex metallic alloys, that have connections to
quasiperiodic phases. Thus restrictions for ordinary dislocations, that are imposed by
the large unit cells, can be overcome.
References
Amazit, Y., Fischer, M., Perrin, B., and Zarembowitch, A., 1995, Proccedings of the
5th International Conference on Quasicrystals, edited by C. Janot and R. Mossieri (Singapore:
World Scientific), p.584.
Beraha, L., Duneau, M., Klein, H., and Audier, M., 1997, Phil. Mag. A, 76, 587.
Bohsung, J., and Trebin, H.-R., 1989, Introduction to the Mathematics of Quasicrystals,
edited by M.V. Jarić (London: Academic Press), p.183.
Boudard, M., Klein, H., de Boissieu, M., Audier, M., and Vincent, H., 1996, Phil.
Mag. A, 74, 939.
Duneau, M., and Audier, M., 1994, Lectures on quasicrystals, edited by F. Hippert and
D. Gratias (Les Ulis: Les Editions de Physique), p. 283.
Edagawa, K., 2001, Mater. Sci. Eng., A309, 528.
Feuerbacher, M., and Caillard, D., 2004, Acta Mater., 52, 1297.
Feuerbacher, M., Metzmacher, C., Wollgarten, M., Urban K., Baufeld, B.,
Bartsch, M., and Messerschmidt, U., 1997, Mater. Sci. Eng., A233, 103.
Frenkel, D.M., Henley, C.H., and Siggia, E.D., 1986, Phys. Rev. B, 34, 3649.
Gratias, D., Katz, A., and Quiquandon, M., 1995, J. Phys. Cond. Matter, 7, 9101.
Katz, A. and Duneau, M., 1986, J. Phys. France, 47, 181.
Katz, A. and Gratias, D., 1994, Lectures on quasicrystals, edited by F. Hippert and D. Gratias
(Les Ulis: Les Editions de Physique), p. 187.
Klein, H., 1997, PhD Thesis, Institut National Polytechnique de Grenoble, France.
Klein, H., Audier, M., Boudard, M., de Boissieu, M., Beraha, L., and Duneau, M.,
1996, Phil. Mag. A, 73, 309.
Klein, H., Durand-Charre, M., and Audier, M., 2000, J. All. Comp., 296, 128.
Klein, H., and Feuerbacher, M., 2003, Phil. Mag., 83, 4103.
Klein, H., Feuerbacher, M., Schall, P., and Urban, K., 1999, Phys. Rev. Lett., 82, 3468.
Koschella, U., Gähler, F., Roth, J., and Trebin, H.-R., 2002, J. All. Comp., 342, 287.
Létoublon, A., de Boissieu, M., Boudard, M., Mancini, L., Gastaldi, J., Hennion, B.,
Caudron, R., and Bellissent, R., 2001, Phil. Mag. Lett., 81, 273.
Mompiou, F., Caillard, D., and Feuerbacher, M., 2004, Phil. Mag., 84, 2777.
Rosenfeld, R., Feuerbacher, M., Baufeld, B., Bartsch, M., Wollgarten, M.,
Hanke, G., Beyss, M., Messerschmidt, U., and Urban, K., 1995, Phil. Mag. Lett.,
72, 375.
Socolar, J.E.S., Lubensky, T.C., and Steinhardt, P.J., 1986, Phys. Rev. B, 34, 3345.
Sun, W. and Hiraga, K., 1996, Phil. Mag. A, 73, 951.
Tsai, A.P., Yokoyama, Y., Inoue, A., and Masumoto, T., 1991, J. Mater. Res., 6, 2646.
	Introduction
	Geometrical models for i-Al-Pd-Mn and its approximants
	Sixdimensional cut-method for the i-phase and approximants
	Rhombic substitution tiling
	Atomic hypervolumes for a geometrical description in hyperspace
	Three- and five-dimensional hyperspace for approximants of Al-Pd-Mn
	Relationships of the hyperspaces
	Phasonic degrees of freedom
	Restriction to excitable degrees of freedom
	Phasonic phase boundaries
	Dislocations: Energetic consideration
	Dislocations in the 'n-phases
	Metadislocations in the three-dimensional model
	Metadislocations in the five-dimensional model
	Metadislocations in the '-phase
	Conclusion
	References

  The approximants xi, xi' and xi'_n of the quasicrystal Al-Mn-Pd display most
interesting plastic properties as for example phason-induced deformation
processes (Klein, H., Audier, M., Boudard, M., de Boissieu, M., Beraha, L., and
Duneau, M., 1996, Phil. Mag. A, 73, 309.) or metadislocations (Klein, H.,
Feuerbacher, M., Schall, P., and Urban, K., 1999, Phys. Rev. Lett., 82, 3468.).
Here we demonstrate that the phases and their deformed or defected states can
be described by a simple projection formalism in three-dimensional space - not
as usual in four to six dimensions. With the method we can interpret
microstructures observed with electron microscopy as phasonic phase boundaries.
Furthermore we determine the metadislocations of lowest energy and relate them
uniquely to experimentally observed ones. Since moving metadislocations in the
xi'-phase can create new phason-planes, we suggest a dislocation induced phase
transition from xi' to xi'_n. The methods developed in this paper can as well
be used for various other complex metallic alloys.

Introduction
A large number of periodic and quasiperiodic phases have been observed in the ternary
Al-Pd-Mn system (Klein, Durand, and Audier 2000). Apart from the stable icosahedral
phase (i-phase) there is a stable decagonal phase (d-phase) with 1.2 nm periodicity and a
metastable d-phase with 1.6 nm periodicity, found by Tsai et al. (1991). The orthorhombic
Ξ-approximants (ξ, ξ′ and ξ′n) of the i-phase form a class of closely related periodic phases
and can be viewed as approximants of this 1.6 nm d-phase, as they have the following
features in common with the d-phase:
(a) They are arrangements of columns of Mackay-type clusters and intermediary atoms
(Sun and Hiraga 1996). These clusters consist of about 52 atoms, placed on concentric
shells of icosahedral symmetry as shown for the ξ′-phase by Boudard et al. (1996).
(b) The columns are in registry, i.e. the compounds also have a layer structure.
The clusters contain about 80% of the atoms. Therefore a coarsened structural de-
scription makes sense where only the projections of the cluster columns along the column
lines are marked. They lead to two-dimensional tilings, which are characteristic for the
respective approximant phase. The vertices of the tiles correspond to the projections of
the cluster columns.
The tilings contain flattend hexagons, which for the ξ-phase are aligned parallel (fig-
ure 1(a)), for the ξ′-phase are staggered in two orientations (figure 1(b)). In the ξ′-phase
isolated combinations of a pentagon and a banana-shaped nonagon (figure 1(c)) are ob-
served, along which the orientation of the hexagons is inverted and which are able to move
by flips. Klein et al. (1996) therefore have termed them phason-lines, following the nota-
tion of related defects in quasicrystals. In bending experiments these phason-lines order
into periodically aligned phason-planes with 1, 2, . . . , n rows of hexagons in between (fig-
ure 1(d) and (e)), forming periodic superstructures. Here we propose to name these phases
ξ′2-, ξ
3-, . . ., ξ
n+1-phases, the reason for the index-shift being given later. Most observed is
the ξ′2-phase, which is also known as Ψ-phase (Klein 1997).
The ξ′n-phases can be considered striped defect lattices. The stripes, which are the
phason-planes, can bend and move, varying their distances (Beraha et al. 1997). Klein
et al. (1999) have observed dislocations in the stripe pattern and have called them metadis-
locations. They are the characteristic textures of partial dislocations in the basic tiling.
In this article we are developing a simplified projection formalism to describe all the
Ξ-phases. It is derived from the hyperspace method of the quasiperiodic phases. As a
consequence, phasonic degrees of freedom can exist in the Ξ-phases which allow move-
ments of the phason-planes. We determine those hyperlattice Burgers vectors which label
the partial dislocations of lowest energies and relate them to the Burgers vectors of the
metadislocations. Indeed, these are the only ones which are observed. The charming fea-
ture of our formalism is that the simplest model is working in three space so that all steps
are easily imaginable.
2 Geometrical models for i-Al-Pd-Mn and its approx-
imants
The atomic positions for a quasicrystal can be described by decorating the lattice points of
a periodic hypercrystal with atomic surfaces and marking the points where these are inter-
sected by a planar cut space (cut-method). An equivalent method is the strip-method (Katz
and Duneau 1986). Approximants are constructed by using inclined cut spaces (Duneau
and Audier 1994). For a detailed explanation see also Gratias, Katz, and Quiquandon
(1995).
0.78 nm
Figure 1: Tilings of four different approximant phases of the 1.6 nm d-phase. (a) Parallel
alignment of hexagons in the ξ-phase. (b) Staggered arrangement of hexagons in the ξ′-
phase. Two hexagon rows are marked grey. (c) The hexagons H occur in two different
orientations. The other tiles, the pentagon P and the nonagon N can only be observed in
combination, called phason-line. The edge length of the tiles is: t6D = 0.78 nm. (d) In the
ξ2-phase there is one row of hexagons between two phason-lines, also called phason-planes.
(e) In the ξ3-phase there are two rows of hexagons between neighbouring phason-planes.
The arrangement of hexagons is flipped at a phason-line as highlighted in the figure.
2.1 Sixdimensional cut-method for the i-phase and approximants
A six-dimensional hyperspace with orthogonal basis vectors e6D1 , . . ., e
6 of length l
0.645 nm was used by Katz and Gratias (1994) to model the i-phase of Al-Pd-Mn. The
hyperspace was decorated with three different atomic surfaces positioned on different nodes
in a face-centred lattice, namely
• even nodes: n0 = {(z1, . . . , z6), zi ∈ Z |
i zi = 0 mod 2},
• odd nodes: n1 = {(z1, . . . , z6), zi ∈ Z |
i zi = 1 mod 2},
• even body-centre nodes: bc0 = {(z1 + 12 , . . . , z6 +
), zi ∈ Z |
i zi = 0 mod 2}.
The cut space is spanned by three vectors, whose components in the above basis are
(τ = 1
5 + 1) is the golden mean):
a6Di = (1, τ, 0,−1, τ, 0), b6Di = (τ, 0, 1, τ, 0,−1), c6Di = (0, 1, τ, 0,−1, τ). (1)
Beraha et al. (1997) applied the cut-method for the three-dimensional atomistic descrip-
tion of the phases ξ and ξ′ as approximants of the icosahedral phase. In order to position
each single atom correctly, they had to modify the atomic surfaces slightly. It turned out
that the centre volumes of the n0 atomic surfaces, which lie again on a six-dimensional
face-centred lattice, correspond to the cluster centre atoms. This way the model is reduced
to three-dimensional tilings for the cluster positions only.
For the ξ′-phase Beraha et al. have derived the vectors that span a unit cell of the
inclined cut space. We add the vectors for the inclined cut spaces of the ξ-phase and the
ξ′n-phases:
a6Dξ = (0, 1, 1,−1, 0, 1), b6Dξ =
(5, 1, 1, 1, 1,−1), c6Dξ = (0, 0, 1, 1,−1, 1) (2a)
a6Dξ′ = (0, 2, 1,−2, 1, 2), b6Dξ′ = b6Dξ , c6Dξ′ = (0, 0, 1, 1,−1, 1) (2b)
a6Dξ′n = (0, 2, 1,−2, 1, 2), b
= b6Dξ , c
= (0, 0, 2n+ 1, 2n,−2n− 1, 2n) (2c)
The b6Dξ -vector is the same for all the phases, because it marks the periodicity in the
tenfold direction of the d-phase, which coincides with the column line of the Mackay-type
clusters. By projecting in direction of b6Dξ , two-dimensional tilings like those in figure 1
can be obtained. The edge length of the tiles is t6D = 1
τ + 2 l6D = 0.78 nm. It can
be calculated by projecting all connection vectors e6Di + e
j of neighbouring n0-sites onto
the tiling plane. The shortest projections have length t6D.
Since the six vectors a6Dξ , c
ξ , a
ξ′ , c
ξ′ , a
and c6Dξ′n = 2nc
ξ′ − a6Dξ′ + 2a6Dξ lie in a
three-dimensional subspace of the six-dimensional hyperspace, a description of the tilings in
a three-dimensional hyperspace is possible by the cut-method. The model can be simplified
by using a simple cubic lattice in three-space, as will be shown in detail in section 2.4. We
can arrive at this conclusion directly after substituting the tiling of hexagons, pentagons
and nonagons by a tiling of rhombs.
2.2 Rhombic substitution tiling
The new tiles are the thin and thick Penrose rhombs which in general are spanned by the
vectors of a regular five-star. The interior angles of the tiles are multiples of 36◦.
As shown in figure 2, a hexagon of the original tiling is substituted by a thick rhomb,
and a nonagon/pentagon combination is substituted by a combination of a thin and a thick
rhomb.1 Therefore we will refer to the thin rhombs as phason-lines. A row of hexagons in
staggered orientation is substituted by a row of alternating thick rhombs, and a phason-
plane is substituted by a combination of a row of alternating thick rhombs and a row of
thin rhombs, which again will be called phason-plane. Hence the number of rows of thick
rhombs between two phason-planes in the new tiling for the ξ′n-phase is exactly n.
Thus the tilings for the column positions of the Ξ-phases emerge as approximants of
the Penrose tiling. However, only rhombs of three orientations show up. To span these
only three prongs of the five-star are required. This is another argument why we can apply
a projection formalism in a three-dimensional hyperspace which was mentioned above and
is elaborated further below.
To model the lattices of all Ξ-phases, their phasonic degrees of freedom and their
metadislocations we found it suitable to resort to a modified cut and projection method.
To our knowledge it has not been applied yet in the literature. It makes use of atomic
hypervolumes and requires a short section.
2.3 Atomic hypervolumes for a geometrical description in hyper-
space
The method will be explained by the example of the well known one-dimensional Fibonacci-
chain: The two-dimensional hyperspace is partitioned into equal unit cells, called atomic
volumes (in general atomic hypervolumes) as shown in figure 3(a).2 For the construction
of an approximant, two different lines are needed: the cut line E (in general cut plane or
cut space) and the projection line E (in general projection plane or projection space, also
named physical space). Those cells that are cut by E are selected (marked grey). The
middle point of those cells is projected onto the projection line.3 The projection leads to
two different intervals on E, either a small one (S), when the neighbouring selected cells
meet vertically, or a large one (L), when they meet horizontally, forming the tiles of the
Fibonacci-chain.
1A similar tiling has been presented by Klein et al. (1996). They used thick rhombs to model the ξ- and
ξ′-phase and small hexagons for the phason-lines. However no description in hyperspace was presented,
explaining the arrangement of the tiles.
2As a generalisation several overlapping atomic volumes could be used, as long as the number of times
a point is covered is constant for the hyperspace. This way the closeness condition (Frenckel, Henley, and
Siggia 1986) of the original cut-method is automatically fulfilled. But we will only make use of the method
of atomic volumes for the canonical case of a hyper-cubic lattice Zn and the unit cells as atomic volumes.
3Any other choice is possible too, as long as it is the same point for each cell, since a different selection
only leads to a global translation of the tiling.
1.26 nm
(c) (d)
(b)(a)
Figure 2: New tilings can be created by substituting the original tiles with thin Penrose
rhombs RN, called phason-lines, and thick Penrose rhombs RB. The ξ-phase (a) and the
ξ′-phase (b) are built only with the thick rhombs, while for the ξ′2-phase (c) and the ξ
phase (d) also the thin rhombs are needed. Between two phason-planes in the ξ′n-phase
there are n rows of alternating thick rhombs (marked grey). The new tiles are shown in (e).
The edge length of the tiles is: t5D = 1.26 nm.
The orientation of E determines the shape of the tiles. In the example of the Fibonacci-
chain, it determines the ratio of S to L. In contrast, the orientation of E fixes the arrange-
ment of the tiles. Since different approximants of the same quasiperiodic phase are built
from the same tiles, but with different arrangement of the tiles, the only difference in the
construction is the varying orientation of E . A rational slope of the cut line leads to a
periodic tiling, while an irrational slope generates a quasiperiodic tiling. The special case
of E and E being identical is the original cut-method as mentioned above.
A geometrical restriction on the orientation of E is imposed by the fact, that projected
tiles must not overlap. In the example of the Fibonacci-chain this means, that the slope
of E must be positive. For a counterexample when it is not fulfilled see figure 3 (b). With
this method we now construct the tilings of the cluster projections.
In the original cut-method the shape of the atomic surfaces depends on the orientation
of the cut space. For the Fibonacci-chain the atomic surfaces are the projections of the
unit cells on the orthogonal complement of E . Because we will deal with cut spaces with
spatially varying orientation in this paper, we would need a varying shape of the atomic
surfaces as well. The method of atomic hypervolumes avoids these problems, as the atomic
hypervolumes themselves do not vary.
2.4 Three- and five-dimensional hyperspace for approximants of
Al-Pd-Mn
Because the rhombic tilings for the Ξ-phases are approximants of the Penrose tiling they
can be described in a five-dimensional hyperspace with basis {e5D1 , . . . , e5D5 }, the hyper-
cubic lattice Z5 and the unit cells as atomic hypervolumes. The projection plane of the
Penrose tiling is spanned by the vectors
a5Dp =
sin(2π 0
), sin(2π 1
), sin(2π 2
), sin(2π 3
), sin(2π 4
c5Dp =
cos(2π 0
), cos(2π 1
), cos(2π 2
), cos(2π 3
), cos(2π 4
and the projection matrix is obtained by writing these vectors normalised in the rows:
π5D‖ =
sin(2π 0
) sin(2π 1
) sin(2π 2
) sin(2π 3
) sin(2π 4
cos(2π 0
) cos(2π 1
) cos(2π 2
) cos(2π 3
) cos(2π 4
. (4)
The projections f i = π
i of the five-dimensional basis vectors give a regular five-star
as drawn in figure 4(a). The rhomb tiles are the projection of two-dimensional faces of the
five-dimensional unit cells. Each is spanned by two vectors of the five-star.
The edge length of the rhomb tiles can be calculated from the edge length of the
pentagon/hexagon/nonagon-tiles as shown in figure 2(e): t6D = 2 cos(72◦) t5D = τ t5D.
This gives: t5D = 1
τ + 2l6D = 1.26 nm, leading to the length of the basis vectors
in the five-dimensional hyperspace: l5D = τ
τ + 2 l6D = 1.99 nm.
A basis {a5D, c5D} for the cut plane of a periodic approximant can be determined with
a basis of the unit cell of the tiling. This is done in figures 4(b)-(e) for various approximant
Figure 3: (a) Construction of an approximant of the Fibonacci-chain by the method of
atomic volumes. A cut line E and a projection line E are required (see text). (b) If the
slope of E is negative, the projected tiles overlap. This case is geometrically not allowed.
In the bottom left and top right corner a small part of the quasiperiodic Fibonacci-chain
is drawn.
(e)(a) (d)
f 3f 4
f5 − f 2
4f 1 + f 2 + f 5
f5 − f 2
6f 1 + f 2 + f 5
f5 − f 2
Figure 4: (a) The projected basis vectors f i lie on a regular five-star. The three rhombs of
the tilings in figure 2 can be constructed with f 1, f 2 and f 5. Unit cells of approximants
and their bases are shown: the ξ-phase (b), the ξ′-phase (c), the ξ′2-phase (d) and the
ξ′3-phase (e).
phases:
a5Dξ = (0, 0, 0, 0, 1), c
ξ = (1, 0, 0, 0, 0), (5a)
a5Dξ′ = (0,−1, 0, 0, 1), c5Dξ′ = (1, 0, 0, 0, 0), (5b)
a5Dξ′n = (0,−1, 0, 0, 1), c
= (2n, 1, 0, 0, 1). (5c)
We now want to derive the transition matrix T 6D5D from the five- to the six-dimensional
hyperspace which maps the basis vectors a5D and c5D on a6D and c6D. The first column
of T 6D5D is fixed by the relation c
ξ = T
ξ . The other columns follow from symmetry
considerations. The fivefold rotation of the hyperspaces that fixes the projection plane is
for the five- and six-dimensional model:
R5D =

0 1 0 0 0
0 0 1 0 0
0 0 0 1 0
0 0 0 0 1
1 0 0 0 0
 , R6D =

1 0 0 0 0 0
0 0 1 0 0 0
0 0 0 1 0 0
0 0 0 0 0 −1
0 1 0 0 0 0
0 0 0 0 −1 0
 . (6)
Observe, that R6D does not change the vector b6Dξ . The symmetry operation in the two
hyperspace must match: R6DT 6D5D = T
5D. This determines the (i + 1)-th column of
T 6D5D via
c6Dξ = [R
6D]−i ◦ T 6D5D ◦ [R5D]i c5Dξ (7)
resulting in:
T 6D5D =

0 0 0 0 0
0 −1 −1 1 1
1 0 −1 −1 1
1 1 0 −1 −1
−1 −1 1 1 0
1 −1 −1 0 1
 . (8)
After fixing a coordinate system for the atomic hypervolumes in the five-dimensional
hyperspace, the approximant phases can appear in five different orientations corresponding
to a cyclic permutation of the basis vectors. Each orientation of the phases ξ′ and ξ′n is
characterised by the orientation of the thin rhombs. By fixing their orientation, a lower-
dimensional hyperspace can be used. Since the third and fourth components of the basis
vectors in (5) vanish, a description in a three-dimensional hyperspace with cubic lattice Z3
(lattice parameter l5D = l3D) and the unit cell as atomic volume is possible by omitting
these components. The transition matrix is simply:
T 5D3D =

1 0 0
0 1 0
0 0 0
0 0 0
0 0 1
 . (9)
For the Ξ-phases this three-dimensional model leads to the same tilings as the five-dimensional
model, but only one orientation of the thin rhombs and two orientations of the thick rhombs
can show up. The projection matrix π3D‖ is derived from π
‖ again by omitting the third
and fourth components:4
π3D‖ =
sin(2π 0
) sin(2π 1
) sin(2π 4
cos(2π 0
) cos(2π 1
) cos(2π 4
2 + τ −1
2 + τ
τ−1 1
. (10)
Bases for the cut planes in the three-dimensional model are given by
a3Dξ = (0, 0, 1), c
ξ = (1, 0, 0), (11a)
a3Dξ′ = (0,−1, 1), c3Dξ′ = (1, 0, 0), (11b)
a3Dξ′n = (0,−1, 1), c
= (2n, 1, 1). (11c)
2.5 Relationships of the hyperspaces
So far the three- and five-dimensional hyperspace have been introduced only phenomeno-
logically for the description of tilings of the approximant phases. Now we want to show
4This projection is not orthogonal, while the projection π5D‖ is orthogonal. That is a consequence of
the omission of two hyperspace dimensions.
how they are related to the original six-dimensional hyperspace. This is meant as a clari-
fication of hyperspace geometry. Mathematically, the formation of the approximants from
the icosahedral quasicrystal proceeds in two steps:
(i) The configuration in direction of a fivefold rotation axis b is changed, while the
configuration perpendicular to this direction in the plane spanned by a and c is
unaltered, leading to a pentagonal or decagonal quasicrystal. This step cannot be
described simply by a reorientation of the cutplane. Additionally a relaxation of the
atoms in direction of b is necessary as shown by Beraha et al. (1997).
(ii) The configuration in the plane spanned by a and c is changed, while the configuration
in direction of b is unaltered. This step is accurately described by a reorientation of
the cutplane as used in this paper.
In step (i) a fivefold symmetry of the icosahedral quasicrystal is conserved. The fivefold
rotation R6D (equation (6)) operates trivially on a two-dimensional subspace U1 spanned
by (1, 0, 0, 0, 0, 0), (0, 1, 1, 1, 1,−1) and as a true rotation on the four-dimensional orthogo-
nal complement U2 spanned by (0, 0, 1, 1,−1, 1), (0,−1, 0, 1,−1,−1), (0,−1,−1, 0, 1,−1),
(0, 1,−1,−1, 1, 0). So for quasicrystals and approximants the lattice vector b is projected
from U1, and the vectors a and c are projected from U2.
The restriction of Z6 onto U2 leads to a non-cubic four-dimensional lattice (the A4
root lattice) that could in principle be used for the construction of the Penrose tiling
and its approximant tilings. By adding a one-dimensional complementary space ∆, these
tilings can be constructed from the Z5-lattice. This is well-known and the usual way to
describe the Penrose lattice. The relation of the five-dimensional hyperspace to the three-
dimensional hyperspace is described in section 2.4. Figure 5 summarizes all the hyperspace
relationships.
3 Phasonic degrees of freedom
Besides standard elastic (phononic) degrees of freedom, quasicrystals have additional de-
grees of freedom (Socolar, Lubensky, and Steinhardt 1986), which originate from the fact
that the cut space is embedded in hyperspace. Local excitations of these so-called pha-
sonic degrees of freedom, correspond to continuous displacements of the cut space along
the direction orthogonal to it in hyperspace. The direction of the excitation is understood
as the direction of the displacement.
The number of the phasonic degrees of freedom for a n-dimensional hyperspace and a
d-dimensional cut space is equal to n− d. As will be shown, phasonic degrees of freedom
are also possible for approximant phases, although then they are not any more continuous
degrees of freedom. For convenience we nonetheless continue using this notation.
U1 ⊕ U2 (6D)
U1 (2D) U2 (4D) U2 ⊕∆ (5D) 3D
Span(b) (1D) Span(a, c) (2D)
Span(a, b, c) (3D)
T 6D5D
T 5D3D
hyperspaces
physical spaces
Figure 5: Relationship of the hyperspaces and the physical spaces. The five-dimensional
space is built on top of the four-dimensional space U2 perpendicular to the fivefold two-
dimensional plane U1. It is not a (full) subspace of the original six-dimensional space. Solid
arrows indicate projections (subspace relationship). Dashed arrows indicate inclusions.
T 6D5D |U2 is an inclusion and T 6D5D |∆ = 0 the zero mapping.
Figure 6: Local excitation of the phasonic degree of freedom for an approximant of the
Fibonacci-chain. The excitation rearranges the sequence of the tiles.
3.1 Restriction to excitable degrees of freedom
In the example of the Fibonacci-chain, a local excitation of the phasonic degree of freedom
is shown in figure 6. Because of the displacement the cut line E now selects different atomic
volumes. In the tiling this leads to a local rearrangement of the tiles, called phasonic jumps.
In general the excitations are considered small and distributed over a wide tiling region,
so that the orientational deviation from the planar cut space is small.
A special case has to be examined separately: If the cut plane of an approximant is
parallel to boundaries of atomic volumes, it is possible that a local excitation of phasonic
degrees of freedom leads to overlapping tiles. If, for example in the Fibonacci-chain, the cut
line of the approximant has slope 0, a local excitation always creates regions of the cut line
having positive as well as negative slope. The regions of negative slope lead to overlapping
tiles, as shown in figure 3(b) earlier. In such a case, accordingly, local displacements of the
cut plane are geometrically not allowed and the phasonic degree of freedom is termed not
excitable.
With regard to the approximants of the 1.6 nm d-(Al-Pd-Mn) in the five-dimensional
model, the phasonic degrees of freedom of the Ξ-phases are not excitable in the directions
of e5D3 and e
4 . So for a full description of these phases the three-dimensional hyperspace
with basis {e3D1 = e5D1 , e3D2 = e5D2 , e3D3 = e5D5 } and only one phasonic degree of freedom is
sufficient.
This last phasonic degree of freedom is excitable in the ξ′n-phases, leading to new
phenomena. But it is not excitable in the ξ- and the ξ′-phase. (The cut plane of the ξ′-
phase, for example, is parallel to the e3D1 -axis). So the phases ξ and ξ
′ behave like normal
periodic crystals, having only phononic degrees of freedom.
It is a pleasant aspect of the three-dimensional hyperspace that the construction for-
malism can easily be visualised. Thus in the next sections we can present the cut plane
n1 × n2 n2 (b)
n1 × n2
Figure 7: (a) Two different cut planes with normal vectors n1 and n2. They have the line
g with direction vector n1 × n2 in common. (b) A transition between the two phases is
realised by adjusting the orientation of the cut plane continuously (see text).
for phase boundaries and dislocations as two-dimensional curved surfaces.
3.2 Phasonic phase boundaries
Different approximant phases are characterised by different orientations of the cut space.
As a feature of the phasonic degrees of freedom, a spatially dependent continuous transfor-
mation from one orientation of the cut space to another is possible, leading to a phasonic
phase boundary.
Let two phases be characterised by the normal vectors n1 and n2 of their cut planes
E1 and E2 (figure 7(a)). For calculating the cut plane E of the phasonic phase boundary
we choose the coordinate system in a way, that the line g = E1 ∩E2 with direction n1×n2
runs through the origin. In the limit far away from g on the side of phase 1 the new cut
plane E contains the vector v1 = n1 × (n1 × n2), while on the other side E the vector
v2 = n2 × (n1 × n2). By using a transition function f : R→ [0, 1] with properties
f(x) = 0, lim
f(x) = 1 and f(0) = 1/2 (12)
the cut plane for the phase boundary E as shown in figure 7(b) is parametrised by:
n1 × n2
‖n1 × n2‖
f(t)v2 + (1− f(t))v1
‖f(t)v2 + (1− f(t))v1‖
; s, t ∈ R
. (13)
Klein et al. (1996) have taken a transmission electron micrograph of a phasonic phase
boundary (figure 8(a)), without interpreting it as such. The phason-lines can be seen as
dark contrasts. Phase 1 on the left side is one of the phases ξ(α), 0 ≤ α ≤ 1 with α ≈ 0.8,
by which we denote an intermediate phase of a ξ- and a ξ′-approximant. Its cut plane
is spanned by aξ(α) = (0,−α, 1) and cξ(α) = (1, 0, 0). Special cases are: ξ(0) = ξ and
ξ(1) = ξ′. Phase 2 on the right side is the ξ′2-approximant.
The tiling for this phasonic phase boundary from ξ(0.8) to ξ′2 is calculated as explained
above and shown in figure 8(b). For the transition function we used:
f(x) =
0 , x ≤ 0
arctan(x)
, x > 0.
In the region of the ξ(0.8)-phase (x < 0) the cut plane is flat, since in the ξ(α)-phases the
phasonic degree of freedom is not excitable, just as in the ξ- or the ξ′-phase. In the region
of the ξ′2-phase (x > 0) excitations of the phasonic degree of freedom show as bendings of
the phason-planes.
As a direct consequence of the hyperspace description the orientation of the phasonic
phase boundary is completely determined by the involved approximant phases. On the
other side, by observing this orientation we can identify the approximant phases, labelled
by the parameter α.
4 Dislocations: Energetic consideration
In quasicrystals and approximants dislocations do exist and are characterised by a unique
Burgers vector b, which now is a vector of the hyperlattice. A dislocation is accompanied
by a phonon- and a phason-strain field, due to the phononic part b‖ = π‖b and the
phasonic part b⊥ = π⊥b (π⊥ projects on the orthogonal complement of E), respectively,
of the Burgers vector. If the phasonic part vanishes, the dislocation is a classical one
and can be described without the hyperspace methods. Since the lattice constants of
periodic approximants are large, classical dislocations have gigantic Burgers vectors with
huge phonon-strains. They are energetically unfavourable and will not be observed.
By extending the linear theory of elasticity to hyperspace (approximating the phasonic
degree as continuous), the line energy E of a dislocation grows quadratically with increasing
lengths ‖b‖‖ and ‖b⊥‖.5 Assuming isotropy in the phononic and phasonic part, E can be
expressed as:
E = cphon‖b‖‖2 + cphas‖b⊥‖2 + ccoupl‖b‖‖‖b⊥‖. (15)
Besides a phononic contribution with elastic constant cphon and a phasonic contribution
with cphas, a coupling term is present with ccoupl.
For estimating the different contributions in (15) the elastic constants in the approx-
imant phases are assumed to be comparable to those of i-Al-Pd-Mn. For the icosahedral
phase the phononic elastic constants have been determined by Amazit et al. (1995) from
sound-propagation. The phasonic elastic constants have been measured with neutron and
x-ray scattering experiments by Letoublon et al. (2001). From the experimental values it
5There is an ongoing discussion whether the elastic energy grows linear (locked state) or quadratically
(unlocked state) in the phasonic strain, see the review by Edagawa (2001). However at higher temperatures,
when dislocations can form and move, the unlocked state is entropically stabilised.
6In several papers the coupling term is neglected, although its appearance is a consequence of the linear
theory of elasticity.
Figure 8: Phasonic phase boundary between the phase ξ(0.8) on the left side and the phase
ξ′2 on the right side. (a) Transmission electron micrograph. The darker spots correspond
to phason-lines (Klein et al. 1996). (b) Calculated tiling for the three-dimensional model
(see text).
can be concluded, that cphas is a few orders of magnitudes smaller than cphon. Furthermore
computer simulations (Koschella et al. 2002) suggest that ccoupl has about the same value
as cphas. So the largest contribution to E comes from the phononic part. In the tiling
picture this means that it is energetically more favourable to rearrange parts of the tiling
than to deform tiles.
5 Dislocations in the ξ′n-phases
The construction method for a dislocation in hyperspace has been discussed e.g. by Bohsung
and Trebin (1989). A dislocation with Burgers vector b forces upon the cut plane a dis-
placement field u with a phononic and a phasonic part. In a first approximation the
displacement field can be distributed isotropically in physical space and becomes in spher-
ical coordinates:
u(r, ϕ) =
b. (16)
5.1 Metadislocations in the three-dimensional model
The steps involved in the construction of a dislocation are visualised in figure 9. The
distorted cut plane has a strain jump of b3D along a radial line as shown in figure 9(a).
The middle points of the atomic volumes selected by the cut plane form a staircase surface
as shown in figure 9(b). This surface is projected with π3D‖ onto the projection plane
(figure 9(c)). Up to now only the phason-strain of the dislocation has been taken regard of
and results in a rearrangement of the tiles. The gap in the tiling is caused by the jump line
in the cut plane. In a last step the phonon-strain is introduced (figures 9(d), (e)), closing
the gap and deforming the tiles. In (figure 9(f)) a Burgers circuit is constructed. The sides
of the tiles correspond to projected basis vectors f i of the hyperlattice. In the example
the lines A, C and B, D cancel each other, only E is left and is lifted to the Burgers vector
b3D = (−2, 1,−3) = −2e3D1 + e3D2 − 3e3D3 .
Consider now an arbitrary dislocation with b3D = (b1, b2, b3), bi ∈ Z. Its phononic
component
b‖ = π
τ + 2(b2 − b3)
τ−1(b2 + b3)
. (17)
must be small for low dislocation energy, as shown in the last section. This is achieved,
if b2 = b3 and
≈ −τ . The best approximating rational values for τ are given by frac-
tions Fm
of successive Fibonacci numbers (Fm)m∈N = (1, 1, 2, 3, 5, 8, 13, 21, . . .) defined
by Fm+1 = Fm−1 + Fm with start values F1 = F2 = 1. They have the following properties,
which can be proved by induction:
(i) Fmτ + Fm−1 = τ
m (18a)
(ii) Fm+1 − Fmτ = (−τ)−m (18b)
(iii) Fm =
5(τm − (−τ)−m) (18c)
(b) (c)
(d) (e)
3f2 3f3
b3D = (−2, 1,−3)
Figure 9: Construction of a dislocation in the three-dimensional model. (a) The displaced
cut plane E of the ξ′3-phase around the dislocation core. (b) Staircase surface of those atomic
volumes, that are cut by E . (c) Projection of the staircase surface onto the projection plane
E. (d) The new tiling around the dislocation core with rearrangements of the tiles due to
phason-strain. (e) The gap in the tiling is closed by phonon-strain. (f) The Burgers vector
b3d can be obtained by performing a Burgers circuit.
Therefore the best candidates for Burgers vectors of low energy dislocations are
b3D =
 Fm−1−Fm
 with b‖ = 1
Fm−1 − τ−1Fm
and length:
‖b‖‖ = τ−m
10 l3D = τ−m · 1.26 nm. (20)
With (8) and (9) we get the 6D Burgers vector:
b6D = T 6D5D T
3D = (0, 0,−Fm−2, Fm−1, Fm−2, Fm−1). (21)
The phasonic component for the six-dimensional hyperspace can be calculated with (18):
‖b⊥‖ =
‖b6D‖2 − ‖b‖‖2 =
2 [(Fm−1)2 + (Fm−2)2] (l6D)2 −
τ−2m(l3D)2
= τm−3
10 l3D = τm−3 · 1.26 nm, (22)
yielding the strain accommodation parameter ζ, i.e. the ratio of the phasonic and the
phononic component of the Burgers vector (Feuerbacher et al. 1997):
= τ 2m−3. (23)
Tilings for dislocations with m = 4 and m = 5 are shown in figures 10(a) and (c). The
dislocation cores appear as triangles which can be used for Burgers circuits. Phason-planes
come in from the left side of the tiling, ending at the flat side of the triangle, their number
being equal to that of the tiles there, which is 2Fm. Whereas the phononic part of the
Burgers vector is much smaller than the side length l3D of the rhombs, the phasonic part
is so large that it leads to a bending of the phason-planes over a huge area of the tiling.
The arrangement of the phason-lines can be viewed as a metastructure which, in the
simple case of a dislocation free ξ′n-phase is a striped centred rectangular lattice. A par-
tial dislocation in the tiling leads to a dislocation in the striped metastructure, named
metadislocation. A dislocation with a three-dimensional Burgers vector as in (19) is called
metadislocation of type m. In contrary to the dislocation constructed in hyperspace, a
metadislocation is an ordinary dislocation in the metastructure with a two-dimensional
Burgers vector.
The Burgers vector bmeta of a type m metadislocation is oriented in the direction of
the lattice vector cξ′n = π
. Since a lattice cell contains two phason-planes, a type m
metadislocation has Fm inserted rows of lattice cells in this direction, leading to a Burgers
vector
bmeta = Fmπ
2n+ τ−1
of length
‖bmeta‖ = l3DFm(2n+ τ−1) = Fm(2n+ τ−1) · 1.26 nm. (25)
The most frequent dislocations are the type 4 metadislocations in the ξ′2-phase. They
have been observed by electron microscopy (Klein et al. 1999), see figure 10(b). From
our theory we derive the lengths ‖b‖‖ = 0.184 nm and ‖b⊥‖ = 2.04 nm (ζ = 11.1)
for the dislocation and ‖bmeta‖ = 17.5 nm for the metadislocation, which is larger by
two orders of magnitude. For comparison we present the metadislocation also in the
hexagon/pentagon/nonagon-tiling (figure 10(d)). Transmission electron micrographs of
type 3, 5 and 6 metadislocations have been published, too, by Klein and Feuerbacher
(2003).
5.2 Metadislocations in the five-dimensional model
Other kinds of metadislocations reported by Klein and Feuerbacher (2003) also show a
large scale arrangement of phason-lines, but require the five-dimensional hyperspace for
description, since phason-lines occur in more than one orientation. Here we will discuss
two different types of metadislocations:
1. Metadislocations with Burgers vector of the form: b5D = (Fm,−Fm−1, Fm, 0, 0): The
phononic component is:
b‖ = π
10(−τ)−m
cos(72◦)
sin(72◦)
. (26)
with length ‖b‖‖ = τ−m · 1.26 nm. The Burgers vector is rotated by 72◦ compared
to the metadislocation in (19). A tiling of such a metadislocation with m = 3 is
shown in figure 11(a). Phason-lines now appear in two different orientations. Fm
rows of tilted phason-lines ending at the dislocation core cannot be explained within
the three-dimensional model.
2. Metadislocations with Burgers vector of the form: b5D = (0, Fm−1, Fm−1, 0, Fm): The
phononic component is:
10(−τ)−m−1
cos(−72◦)
sin(−72◦)
. (27)
with length ‖b‖‖ = τ−m−1 · 1.26 nm. A tiling with m = 5 is shown in figure 11(b).
This time Fm−1 rows of tilted phason-lines end at the dislocation core.
For transmission electron micrographs of the two discussed metadislocations see fig-
ure 11(c) and (d). In the first figure the gap extending to the left in the tiling has been
closed by the motion of the phason-planes. In the second figure the metadislocation seems
to have moved up, while building a sack-shaped area of ξ′-phase below. On the sides of
this area, five phason-planes are cut into halves by a region of ξ′-phase.
(c) (d)
Figure 10: (a) Tiling of a type 4 metadislocation in the ξ′4-phase. (b) Transmission electron
micrograph of a type 4 metadislocation in the ξ′2-phase, courtesy by H. Klein and M.
Feuerbacher. (c) Tiling of a type 5 metadislocation in the ξ′4-phase. (d) For comparison
the calculated hexagon/pentagon/nonagon tiling of a type 4 metadislocation in the ξ′4-
phase is shown.
20b3D‖
20b3D‖
Figure 11: Tilings of metadislocations in the ξ′1-phase with Burgers vectors (a) b
(2,−1, 2, 0, 0) and (b) b5D = (0, 3, 3, 0, 5). The phason-strain field is not distributed
isotropic. The phononic component of the Burgers vector is drawn magnified 20 times,
since it is too small otherwise. (c) and (d) show transmission electron micrographs of these
two metadislocations, courtesy by H. Klein and M. Feuerbacher.
(a) (b)
Figure 12: (a) A tiling of a metadislocation in the ξ′-phase with six inserted phason-planes.
In the dislocation tail a strip of ξ′3-phase is created. (b) Cut plane for the metadislocation
in the ξ′-phase. The phason-strain field is not relaxed isotropic, but concentrated in the
dislocation tail, where the phason-lines are observed.
6 Metadislocations in the ξ′-phase
Although phasonic degrees of freedom are not excitable continuously in the ξ′-phase,
metadislocations with phasonic components, which cause a noncontinuous local phason-
strain do exist and have been observed experimentally by Klein and Feuerbacher (2003).
They can be described in the three-dimensional hyperspace. To minimise the dislocation
energy, their Burgers vectors b3D are the same as in the ξ′n-phases (19). For a tiling see
figure 12(a) in the case m = 4. Similar to metadislocations in the ξ′n-phases, a tail of six in-
serted phason-planes ends at the flat sides of the dislocation core. Towards the dislocation
core the phason-planes converge, while far away they run parallel. Contrary to metadislo-
cations in the ξ′n-phases the phason-strain now is not distributed isotropically around the
dislocation core, but concentrated in the region, where phason-lines are observed.
The shape of the cut plane for the metadislocation in the three-dimensional model is
shown in figure 12(b), where the coordinate system is rotated, so that the cut plane Eξ′ of
the ξ′-phase lies horizontally. Because of its small phononic component, the Burgers vector
is almost perpendicular to Eξ′ .
We place the dislocation in the origin and the strain jump of b3D along the y-axis, along
a line in the middle of the strip of phason lines. Inside the dislocation tail and far away
from the core the strain field is
u3D(x, y) =
|x| −
b3D. (28)
The parameter λ controls the width of the dislocation tail. To simulate a convergence of
the dislocation tail λ = λ(y) must be designed decreasing with distance from the dislocation
core. Outside of the dislocation tail, the cut plane is parallel to Eξ′ . The region with the
phason lines can be viewed as a very thin strip of newly created ξ′n-phase, embedded in
surrounding ξ′-phase. We suggest, that these dislocations move in the direction of their
tail, because for the movement perpendicular to it the entire tail would have to be dragged
along. The movement is pure climb, because it is perpendicular to the burgers vector and
the dislocation line. There are indications (Mompiou, Caillard, and Feuerbacher 2004),
that in the i-Al-Pd-Mn quasicrystal dislocations can move by pure climb. Furthermore
the most often observed dislocations in i-Al-Pd-Mn have the same burgers vectors as the
metadislocations in the Ξ-phases (Rosenfeld et al. 1995).
Finally it should be mentioned that Feuerbacher and Caillard (2004) also observed
other dislocations in the ξ′-phase with Burgers vectors that do not lie perpendicular to
the vector b6Dξ′ . These cannot be pictured as two-dimensional tilings, but require a true
three-dimensional description in physical space.
7 Conclusion
In this paper we have introduced a simple projection formalism in five-dimensional hyper-
space for the description of the i-(Al-Pd-Mn) approximants ξ, ξ′ and ξ′n as two-dimensional
tilings. The tilings correspond to the projections of clusters columns. In most cases the
formalism can even be restricted to a three-dimensional hyperspace. The tilings are gen-
erated by cuts and projections through a hyperspace which is partitioned into atomic
hypervolumes.
We have shown that phasonic degrees of freedom in form of continuous displacements
of the cut space do exist in these phases and can be either excitable or not. They play a
fundamental role for phasonic phase boundaries as well as dislocations. In the case of the
ξ- and ξ′-phase no phasonic degrees of freedom are excitable, while in the ξ′n-phases there
is exactly one excitable phasonic degree of freedom, which is connected to the bending of
the phason-planes.
Nevertheless metadislocations in the ξ′-phase can exist, creating phason-planes and
forming a thin strip of ξ′n-phase in their tails while moving by climb. In this way, a consec-
utive motion of metadislocations through the ξ′-phase induces a phase transformation to
a ξ′n-phase, making the phasonic degree of freedom excitable. The same metadislocations
continue to exist in the ξ′n-phases.
It is possible that phasonic degrees of freedom and dislocations with phasonic com-
ponents also occur in various other complex metallic alloys, that have connections to
quasiperiodic phases. Thus restrictions for ordinary dislocations, that are imposed by
the large unit cells, can be overcome.
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	Introduction
	Geometrical models for i-Al-Pd-Mn and its approximants
	Sixdimensional cut-method for the i-phase and approximants
	Rhombic substitution tiling
	Atomic hypervolumes for a geometrical description in hyperspace
	Three- and five-dimensional hyperspace for approximants of Al-Pd-Mn
	Relationships of the hyperspaces
	Phasonic degrees of freedom
	Restriction to excitable degrees of freedom
	Phasonic phase boundaries
	Dislocations: Energetic consideration
	Dislocations in the 'n-phases
	Metadislocations in the three-dimensional model
	Metadislocations in the five-dimensional model
	Metadislocations in the '-phase
	Conclusion
	References
