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Topology of the phason degree of freedom, phason singularities, and diffusive motion in octagonal
quasicrystals
Anke Trub, Hans-Rainer Trebin
To cite this version:
Anke Trub, Hans-Rainer Trebin. Topology of the phason degree of freedom, phason singularities, and diffusive motion in octagonal quasicrystals. Journal de Physique I, EDP Sciences, 1994, 4 (12), pp.1855-1866. �10.1051/jp1:1994226�. �jpa-00247039�
Classification
Physics Abstracts
61.40M 61.70P 66.30F
Topology of the phason degree of IFeedom, phason singularities,
and diffusive motion in octagonal quasicrystals
Anke Trub and Hans-Rainer Trebin
Institut für Theoretische und Angewandte Physik der Universitàt Stuttgart, Pfaffenwaldring 57, D-70550 Stuttgart, Germany
(Recelvert 29 December 1993, revised 13 May1994, accepted 25 August 1994)
Résumé. Nous étudions deux points particuliers du degré de liberté des phasons: prenlière- ment, nous montrons que l'espace perpendiculaire n'est ni connecté ni simplement connecté et que les défauts ponctuels, linéaires ou plans peuvent donc exister dans les pavages quasi-
cristallins. Deuxièmement, en changeant la variable de phason localement ou globalement, on
peut déplacer les sommets le long d'une boucle ou bien le long des chemins étendus jusqu'à l'infini- Nous donnons des exenlples explicites pour l'autodiffusion assistée par les phasons, qui
a été proposée récemment.
Abstract. The phasou degree of freedom in quasicrystals is studied in two respects: first it is poiuted eut, that the orthogonal space E~ is net connected and Dot simply connected and that therefore wall, line aud point defects exist in the phasou variable of quasicrystalline tilings.
Second, both by global and local changes of the phason variable vertices can be transported either aloug loops or along paths exteudiug to iufiuity. Thus
we provide expirait examples for the recently proposed phasou-nlediated selfdiffusion.
l. Introduction.
Trie X-ray or electron diffraction patterns of quasicrystals display sharp Bragg peaks, which, however, are arranged in a noncrystallographic symmetry. Hence trie mass density can be
superposed by a set of plane waves, consisting of D basic waves of incommensurate wave vectors Ki,
,
KD, and their higher harmonics of wave vectors q = ni Ki +. nDKD, ni E lZ.
D is larger than the dimension d of physical space, and the mass density is quasiperiodic. But it can be represented as a cut through a periodic mass density in a D-dimensional hyperspace
E = E" e E~, which divides into physical space E" and perpendicular space E~. Trie phases
+t~ of trie basic waves g~e~~J ~ with g~ = (g~(e~~J can be changed freely, without altering trie
physical properties [Ii. They can be expressed as
+t~ = K)' tt + K) w with K)' and K) being
@ Les Editions de Physique 1994
projections of vectors of the reciprocal lattice corresponding to E" and E~, and the vectors tt
and w being shifts of the physical plane E". The component tt along E" simply translates the quasicrystal, the component w along E~ causes local structural changes. E~ con be considered
as an order parameter space and the corresponding order parameter w frequently is denoted,
"phason variable" of the quasicrystal.
This article extends previous studies of the phason variable [2-8] by analyzing the changes of quasiperiodic tilings, if one moves along straight lines or closed loops in phase space. The method of atomic surfaces and the strip-projection method are used here. Technical details for their application are repeated in section 2.
For a large class of tilings, a global change of the phason variable makes vertices jump along systems of rows, for which explicit construction rules exist [9, 8]. Inhomogeneous tilings are
characterized by a spatially varying vector w, just as a deformed periodic crystal carries a field of displacement vectors. We point out here explicitly, that for the classification of phason singularities, walls, lines and points have to be taken out of perpendicular space E~. With these reductions E~ is not
any more connected and simply connected and gives rise to wall,
line and point singularities in the phason variable (Sects. 3 and 4).
Once a loop in E~ is traversed, sets of vertices at the intersections of jumping rows are permuted [3]. The composition of several loops leads to hopping motions of vertices along large rings in trie tiling. Extending these rings to infinity we can provide explicit examples for
infinitely extended, percolating self-diffusion patins (Sect. 5).
It has been emphasized by Kalugin and Katz [iii, that phason-mediated hopping of atoms is a new mechanism for self-diffusion- When at a critical amplitude of trie phason amplitude trie percolation patins are being opened, trie diffusion coefficient should increase rapidly up
to a saturation limit. Such a behaviour of trie diffusion coefficient bas only been observed in Monte-Carlo simulations of the octagonal tilings of the plane [12]. Hints for the existence of phason-mediated self-diffusion come from experiments on plastic deformations of quasicrystals.
At low temperatures quasicrystals are brittle, as dislocations create walls of phason-singularities along their paths [13, 14] and, thus, are impeded in their mobility. As soon as the walls con dissolve by self-diffusion, quasicrystals become ductile. One might speculate that the onset of ductility is close to the temperature of the onset of enhanced self-diffusion.
Quasiperiodic patterns can be viewed as limits of periodic patterns with increasing lattice constant. In these rational approximants the phason singularities are stacking faults and can be viewed as "topological semidefects" (Sect. 6).
All these studies are performed primarily on the standard octagonal tiling of the plane (Sect.
2), but we shall comment the extension of the results to other tilings, also in three dimensions.
2. Preliminaries.
Trie octagonal tiling is constructed either by the strip-projection method or, equivalently, by the method of atomic surfaces (Fig. l). There, the twodimensional tiling plane E" is embedded
in a four-dimensional hypercubic lattice lZ~ under irrational slopes. The projection of trie cubic unit cell onto perpendicular space E~ yields a regular octagon with edges parallel to trie
projected vèctors e<~> and of length (e<~>(. A copy of this octagon is attached to trie center of each unit cell, and trie edges of adjacent octagons are connected by pieces of planar surfaces
parallel to E", denoted "steps" [3]. Trie projection of trie orthonormal basis vectors eo, fi, e2, e3 of lIt~ to E~, e<~>, and to E", e~, and an octagonal piece of trie atomic surface are drawn in
figure 2. The intersection points of the atomic surfaces with E" are the vertices of the tiling.
A tiling may be labeled by the vector w E E~ describing the crosspoint of E~ and E", which
is denoted phason variable.
~
E~
Fig. 1. Method of atomic surfaces. The intersections points of the atomic surfaces with E" yield
the vertices of the tiling.
ci cri
e<o>
e<3>
e~
Fig. 2. Projections of the canonical four-dimensional basis vectors enta E~ and E",
an atomic surface and octagonal part of the tiling.
A change in w corresponding to a shift of E" parallel to E~ changes the tiling, because after the shift E" crosses several new octagons and has left others. Theses changes are studied best, when trie octagons are projected into E~. Their midpoints are connected by integer
3
combinations of trie projections e<~> of e~, ~jn~e<~>; trie corresponding vertices in E" are
1=0 3
separated by vectors £
n~e~. Ail surfaces, whose projection contains trie crosspoint w, yield
1=0
vertices of trie tiling.
Given one octagon in E~, a countable but dense set of octagons also exists shifted horizontally
to it by integer combinations of trie incommensurate vectors e<o> and e<3> e<i> (Fig. 2).
Those of the set containing w correspond to the interior vertices of hexagons, which in the marked worm of figure 3 form a quasiperiodic sequence. This sequence can be constructed by
a strip-projection method [8, 9] in a rectangular lattice with base-vectors eo and e3 fi and
Fig 3. Worm of hexagons and squares, where the interior vertices of the hexagons correspond to
points close to the edge of the atomic octagons m E~.
,2
8. .3
7
a) b)
Fig. 4. (a) Eight octagons m E~ sharing a common vertex, and loop around this vertex. (b)
The eight possible positions of vertices mside an octagonal cage correspondmg to the octagons of the previous figure.
with a strip of width (eo + e3 fil Ail the lattice-points inside the strip represent the inner vertices of a worm, having two possible separations, eo ei + e3 and 2eo ei + e3.
Another characteristic configuration in E~ are eight octagons sharing a common vertex
(Fig.4). They correspond to trie eight possible positions for vertices inside an octagonal cage
in the tiling which themselves form an inner octagon. Only three of the eight atomic octagons
in E~ contain w and give use to the three interior vertices of the octagon in E", where four of
the hexagon-square rows are meeting. Those atomic octagons, which are intersected close to an edge, correspond to vertices of the tiling in the interior of a hexagon [8j.
3. Simple changes of trie phason variable and phason~wall singularities.
When w crosses the edge of two adjacent atomic surfaces in E~, such
a hexagon vertex is jump-
ing in E" and is causing a flip of the hexagon. The step connecting the two edges in hyperspace
E marks a topological neighborhood and allows us to follow the hopping path of a vertex. If
w sits precisely on the edge, the position of the vertex inside the hexagon is ambiguous. As
soon as a point or set of points in an order parameter space gives rise to ambiguities, like the origin in the space of magnetization vectors for the direction of magnetization, it has to be removed. Hence ail edges must be taken out of E~, dividing it into disconnected pieces, and simple changes of the phase lead to wall-singularities (in two dimensions to line singularities [loi. If we move across E~ by a distance zlw, then parallel sets of hexagon rows flip, whose sequence again can be determined by trie strip-projection method [8j. When a straight line is drawn across trie plane, on one side of which trie phason variable is w, on trie other w + zlw,
trie line is crossing rows of hexagons, which on one side have not been flipped, on the other are
flipped. Therefore at these intersections phason-singularities or "mistakes" are sitting. A wall
singularity of trie phase is a string of "mistakes".
If w is placed on a common corner of eight octagons in E~, there is an ambiguity inside an
octagonal cage in E". This
corner must be taken out of E~, giving rise to a point singularity,
which is composed of four phason singularities. An example bas been drawn in figure 5 in trie form of a vortex. Once this vortex is entoured in trie tiling, trie phason variable is encircling
the corner in E~ along a small loop (Fig. 4). Defects of this type bave been addressed in
Penrose-tilings as "decapod-defects" [là, 16j.
In three-dimensional quasicrystals, like the Ammann-Penrose-Kramer tiling, common faces, edges and corners of atomic polytopes must be removed from E~, giving rise to wall-, line- and
point-defects in the phase.
o o o o
Fig. 5 Fig. 6
Fig. 5. Vortex phason singularity corresponding to the elementary trop.
Fig. 6. Loop around two adjacent vertices in E~, belonging to 14 octagons.
All these defects are due to inhomogeneous phase-fields +t(x) in the tiling. In the next section
we consider a phason variable constant for trie entire tiling, but changing in time and moving along loops in E~.
4. Phason kinetics.
When w is being changed globally along a loop, encircling the common corner of octagons (Fig.4), the four worms crossing the octagon in E" flip successively forward and backward, causing the three vertices inside the octagonal cage to permute cyclically on the inner octagon each tour [3]. Ourintention is to investigate loops around an arbitrary number and arrangement
of such corners in E~. We look for general rules to divide arbitrary loops into loops around
single corners and try to treat the corresponding permutations of vertices in E".
It is possible to extend the elementary loop by moving along a closed loop containing two
corners m E~, which are separated by a vector e<~>. Fourteen octagons are involved in the
configuration, rive of them being intersected by E" (Fig. 6). The loop in E~
cari be deformed into a loop around trie right vertex, a translation, a loop around trie left vertex, and trie back translation. Trie sequence is depicted in figure 7. In E", first three vertices in trie right octagon are being permuted. Depending on the configuration it may be that some vertices
proceed to the next position during the translation. After the three vertices of trie left octagon bave exchanged their place, trie translational movement is being reversed. As result, after
transversal of trie loop, trie rive vertices bave advanced by one position (Fig.8).
One can generalize this result and move along extended loops by encircling more and more
common sites of atomic octagons. Examples for motions of 5+11
= 16 and 2 x 5+3+2 x1+25
=
40 are presented in figures 9 and 10. In these larger systems trie inner octagons form rings,
1 4
~',
. . , ',
,' ,
i '
, '
j
'~
,
2 3
~j Î
,
,,
,'~~
7 ", ,'
1 '
1 1
' ,
'
, a',,
' 1
~~~' '~
b)
Fig. 7.
arouud the
point, a to the left , the
second
loop,
and a ranslation back. (b)
The
permutations on the inner in E" are erformed the same order. First the vertices
symbolized by an circle on the ight,
then on the left. The result is the cychc pernlutation
vertices.
Fig. 8
Fig. 9
Fig 8 Permutation of rive vertices in E" when in E~ two adjacent vertices are being encircled.
Fig. 9 Permutation of atoms along loops with 5 +11 atoms.
Fig 10. Permutation of atoms along loops with 2 x 5 + 3 + 2 x 1+ 25
= 40 atoms.
and therefore there are separated systems of jumping vertices. In trie case of trie eight octagons trie vertices move on two concentric, ring-shaped patries in opposite directions.
In all these examples we were looking for connected chains of octagons in E". These
corre-
spond to sets of corners in E~ connected by trie basic vectors e<~>, i = 0,1, 2,3.
One must be aware, however, of trie following. to perform changes of trie tiling, where only
a subset of trie vertices is supposed to jump, trie loop in E~ must stay within an area which
is contained in ont octagon. Otherwise each original octagonal surface has been left by w and
no vertex has remained in~ its place.
5. Self-diffusion.
Chains of octagons extending to infinity open up percolating paths for self-diffusion of atoms, enabled by the phason degree of freedom of the quasiperiodic tiling. To detect these chains, Kalugin and Katz ilIi have enlarged the atomic surfaces by a scaliqg factor e. An infinitesimal
e leads to an overlap of the atomic surfaces at common corners. If w is contained in this
overlapping area, ail eight vertices corresponding to the eight surfaces appear as an octagon inside the cage. If e is enlarged still more, many of such octagons arise, forming at last infinitely
extended chains for e
= 1+ 1/v5.
We are interested in an explicit construction of a percolating path which enables self~dioEusion.
We use the same method as described in the previous section and look for motion caused by permutation of vertices in an infinite chain of octagons.
An infinite chair of octagons in E" corresponds to an infinite chair of points p~ in E~, which
are the common corners of eight atomic surfaces and are separated bj basis vectors e<~>. The
position of point p~ therefore is:
3
P~ "
~ n~e<j>
j=0
Because the path connecting ail the points must remain inside an atomic surface, it is
multiply folded. Its image in E", which is formed by the chair of octagons, must
cross the
plane along an infinitely extended path. By means of cyclic permutations of the vertices, we
cari obtain a path, on which the atoms move through the whole quasicrystal.
To construct a suitable set of points (p~) in E~
we observe, that there are collinear but incommensurate vectors like e<o> and e<i> e<3> in E~ They point into opposite directions
(Fig. 2) and cari be attached in a quasiperiodic sequence such that they remain inside an atomic octagon. Their partners eo and ei e3 in E" point into the same, positive direction (Fig. 2),
and the corresponding sequence in E" proceeds into direction eo from minus to plus infinity.
Both sequences can be constructed by a strip-projection method similar to that of figure II
involving a rectangular lattice spanned by eo and fi e3. The strip of width (e~( points in the direction of the straight hne h" being the intersection of E" with the lattice plane, and the points inside determine the sequence of octagons. As in our required sequence only separations by edge vectors are allowed, each step by e<i> e<3> must be composed of two steps by e<i>
first and then -e<3> or reversely.
Vertices now are transported from one octagon to the next as shown in the basic combination process of figure 7: A corner in E~ is encircled, then a translation is performed to the next corner, which is encircled also, and so on. In total a rather comphcated path has to be followed in E~ This path can be unfolded and reduced to a simple loop encircling the entire chains of
corners in E~ (Fig.12). Once this loop has been encircled, atoms of an infinite chain in E" have
jumped by one position. By permanently encircling trie loop, vertices are transported along an
unbounded patin (Fig.13). There is, however, also a partner chain, where trie motion occurs in trie opposite direction. Thus, we bave provided an explicit example for diffusion through
periodic changes of the phase.
The transport as proven above was performed by global changes of the phason variable, which are, however, not necessary. There is a dioEerent process to transport one single atom
~<l>
<3>
Î~ ~<ll>
o o
E~
ig.
Fig.
The traight fines h~
espectivly. The along h" has the width [e~[, the ength of
determines the sequence of
octagons E".
Fig. 12. - Loop in E~, which, when ncircled, leads to tomic transport along a chair of octagons
i~ Ell,
Fig. 13. Along the chain atomic transport to the right is accompanied by to the left
other rows.
along a of
octagons through the whole To obtain for a single atom
we only
in a strip of width 3(ei + e2 + e31 along the
chain. hese single flips
variation
of w. An xample for this rocess is trie imple cyclic exchange of atoms
which is performed by
eight jumps only
nside the cage. It can happen
exclusively ithin the octagon and orresponds to creation of four hason-antiphason pairs
(Fig. 14) and heir successive nihilation. This method can cause the flips of several hexagons
inside the strip along the chain and moves one atom from octagon
these remain existing~ sometimes they annihilate subsequently. As
seen above trie
permutation of three vertices effectively does not produce
phason-antiphason-
pairs. Trie only ones remaining after trie whole
process
from one
octagon to trie next. To move one single atom through
permutating vertices only one needs a
"cloud" of up to ten
phason-antiphason-pairs, most of which
annihilate subsequently. After trie vertex bas passed along trie patin there is an verage
rate of one per left. An xample of a possible patin for
JOURNAL DE PHYSIQUE1 -T 4 N' 12
DECE~IBER lQ94 67
