\documentstyle[aps,preprint,epsfig]{revtex}


\begin{document}

\title{Total-energy-based prediction of a quasicrystal structure}

\author{M. Mihalkovi\v{c}$^{1,2}$, I. Al-Lehyani$^3$, E. Cockayne$^4$,
C.L. Henley$^5$, N. Moghadam$^6$, J.A. Moriarty$^7$,
Y. Wang$^8$, M. Widom$^3$}

\address{
$^1$ Institute fur Physik, Technische Universit\"{a}t Chemnitz, 
D-09107 Germany \\
$^2$ Institute of Physics, Slovak Academy of Sciences, Bratislava, Slovakia \\ 
$^3$ Department of Physics, Carnegie Mellon University, Pittsburgh PA  15213 \\
$^4$ Ceramics Division, NIST, Gaithersburg MD 20899-8520 \\
$^5$ Department of Physics, Cornell University, Ithaca NY  14853 \\
$^6$ Oak Ridge National Laboratory, Oak Ridge TN 37831-6114 \\
$^7$ Lawrence Livermore National Laboratory, Livermore CA 94551 \\
$^8$ Pittsburgh Supercomputer Center, Pittsburgh PA  15213 \\
}
\date{\today}

\maketitle


\begin{abstract}
Quasicrystals are metal alloys whose noncrystallographic symmetries
challenge traditional methods of structure determination.  We employ
quantum-based total-energy calculations to predict the structure of a
decagonal quasicrystal from first principles considerations. Our Monte
Carlo simulations take as input the knowledge that a decagonal phase
occurs in Al-Ni-Co near a given composition, and use a limited amount
of experimental structural data.  The resulting structure obeys a
nearly deterministic decoration of tiles on a hierarchy of length
scales related by powers of $\tau$, the golden mean.
\end{abstract}

\newpage

\section{Introduction}
\label{Intro}

Al-Ni-Co forms thermodynamically stable and highly perfect decagonal
quasicrystalline samples over a range of
compositions~\cite{PhaseDiagram}. Of special interest is the
composition Al$_{0.70}$Ni$_{0.21}$Co$_{0.09}$ for which the structure
is periodic along the z axis with a period of $c=4.08$~\AA, and
quasiperiodic perpendicular to this axis with a characteristic length
(termed a ``quasilattice constant'') of $a_0=2.45$~\AA~\cite{basicNi}.
This ``basic Ni'' composition is well suited for theoretical modeling
because it should be the simplest structure, lacking the quasiperiodic
modulation and $c$-axis doubling observed at other compositions.
Numerous attempts to determine the structure of this compound start
from experimental
data~\cite{Patterson,AlNiCoModels,Z-contrast,Takakura,Cervellino} but
do not {\it predict} a structure on the basis of total energy.

The transition metals Ni and Co (generically denoted ``TM'') play
similar chemical roles in Al-transition metal quasicrystals, and they
are not distinguished by ordinary X-ray or electron diffraction. In
contrast, we distinguish Ni and Co in this work, treating Al-Ni-Co as
a ternary system.

Our work employs total energy calculations based on pair-potentials
derived from first principals electronic structure considerations.
Monte Carlo simulations are used to find favorable atomic decorations
and tiling structures.  The {\em existence} of a 10-fold symmetric
stable phase, and the lattice constants of that phase are taken as
experimental input. The pair potentials, experimental input and Monte
Carlo methods are described in section~\ref{Methods} below.

Using these methods we find low energy structures can be described
using nearly deterministic decorations of rigid tiles. Specific
features of the pair potentials are responsible for certain details of
the structure over short length scales, as we describe in
section~\ref{Local}. Taking decorated tiles as fundamental objects, we
then perform larger scale simulations and find an alternate structural
description (see section~\ref{Supertiling}) in terms of tiles whose
edge length is a factor of $\tau^2$ larger ($\tau=(\sqrt{5}+1)/2$ is
the Golden Mean). Details of our structural model are compared with
experimental data in section~\ref{Comparison}.

\section{Methods}
\label{Methods}

Our total energy calculations employ quantum-based pair-potentials
derived from the generalized pseudopotential theory (GPT)~\cite{GPT}.
GPT expands the total energy in a series of volume-, pair- and
many-body-potentials. The volume term exerts no force and may be
neglected at fixed volume and composition. The many-body terms are
small except among clusters of neighboring transition-metal atoms, and
we incorporate their influence with modified short-ranged TM-TM pair
interactions constrained by full {\em ab initio} calculations~\cite{mGPT}.

Fig.~\ref{pot} displays the resulting pair potentials~\cite{GPT,mGPT}.
Note that they oscillate with a spatial frequency that asymptotically
approaches $2k_F$, with $k_F$ the Fermi wavevector. Compatibility of atomic
separations with minima of the oscillating potentials can lower the total
energy. This can be shown using second order perturbation theory
within the independent electron approximation when $2k_F=|{\bf G}|$,
with ${\bf G}$ a reciprocal lattice vector. This is one of the
Hume-Rothery mechanisms for structural stability. For our simulations we
cut off the potential using a smooth truncation on the interval
from 8 to 10~\AA.

We identify four salient short-ranged properties of the pair
potentials: (1) V$_{\rm AlAl}(r)$ has a broad shoulder starting around
$r = 2.9$~\AA~ and is repulsive at shorter distances; (2) V$_{\rm
AlCo}(r)$ and V$_{\rm AlNi}(r)$ exhibit deep first minima near
$r=2.5$~\AA~ and second minima near $r=4.5$~\AA; (3) The V$_{\rm
AlCo}$ well is significantly deeper than the V$_{\rm AlNi}$ well; (4)
The modified $V_{\rm TMTM}(r)$ have shallow minima near r=2.6~\AA.

The following features of the $d$(AlNiCo) structures are evident in
the experimentally determined Patterson function~\cite{Patterson,Cervellino}
which contains a peak at every interatomic vector $\bf r$: (A) All
atoms lie on or nearly on layers separated by $c/2=2.04$~\AA; (B) The
vector from an atom to a nearest neighbor (with a tolerance of $\sim
0.1$~\AA) belongs to a small, discrete basis set of ``linkage''
vectors; (C) The in-plane components of linkage vectors are $\pm a_0
{\bf e}_i$ (see Fig.~\ref{layer}) or simple sums of such vectors.

We construct trial quasicrystal structures that achieve low total
energy while satisfying the above experimental constraints. To enforce
constraints (B) and (C) we limit atomic positions to a collection
of discrete sites (Fig.~\ref{layer}), located at vertices of a
two-dimensional tiling of rhombi with edge $a_0$ and acute angles
$36^\circ$ or $72^\circ$. To enforce constraint (A) we stack two
independent tilings above each other.  As the tiles can be placed in
many ways, and atoms distributed randomly among the sites, these {\it
minimal constraints} permit a great variety of structures, including
all reasonable quasicrystal structures. After we discover favorable
low energy motifs consistent with the minimal constraints, we
remove unnecessary degrees of freedom, effectively defining
{\it highly constrained} models.

Metropolis Monte Carlo annealing, in which the temperature is reduced
slowly in discrete steps, yields low energy structures. Care is taken
to anneal for long times at those intermediate temperatures where
significant degrees of freedom begin to freeze.  Two kinds of Monte
Carlo steps are employed: (i) swaps between nearby atoms (within
6~\AA) of different species in either layer, including hops of one
atom to an empty ideal site nearby; (ii) ``flips'' (in one layer)
which reshuffle the three rhombi in a fat or thin hexagon (both types
of hexagon are shown in Fig.~\ref{layer}). Due to the hexagon flips,
our ensemble is an ``equilibrium random tiling'' allowing phason
disorder~\cite{Phason}, but the system is free to find a unique state
if that is favored by the potentials. Our simulations are performed
with periodic boundary conditions using cell sizes chosen to best
approximate the quasiperiodic structure. Additional simulation details
are given in section~\ref{Results}. Note that certain Monte Carlo
steps of type (i) (atom swaps) are prohibited in some of our nearly
deterministic decoration simulations.

\section{Results}
\label{Results}

\subsection{Local order from minimally constrained simulation}
\label{Local}

In our initial simulation with minimal constraints, we employ a cell
of size $12.22 \times 14.37 \times 4.08$~\AA$^3$ and composition
Al$_{34}$Ni$_{12}$Co$_{4}$.  This cell contains 72 ideal sites, 36 in
each layer with the $c$-axis periodicity enforced. We cool slowly
starting from a random high energy configuration. Our initial
annealing is carried out at very high temperature ($\beta\equiv 1/k_BT
= 0.5$ eV$^{-1}$, T=23208K), with the temperature dropping eventually
to near room temperature ($\beta=35.0$ eV$^{-1}$, T=332K). This protocol
identifies a unique minimum energy configuration illustrated in
Fig.~\ref{small}a.

The optimal configuration can be described simply in terms of a new
{\it highly constrained} model (see Fig.~\ref{small}b) that obeys the
following rules: (i) the entire plane is tiled by three compound tiles
called ``hexagon'', ``boat'', and ``star'', outlined by heavy edges
and built respectively of three, four, and five rhombi (this is called
the ``HBS'' tiling~\cite{CW}).  (ii) The optimal decoration of the HBS
tiles is virtually unique.  Minimally constrained simulations with
larger cells support these rules.

We can understand this decoration in terms of the salient features of
the potential we enumerated at the start.  In view of features (1) and
(2), the minimum-energy structure must maximize the number of Al-Co
and Al-Ni bonds.  In view of feature (3), every Co ought to have
purely Al neighbors, which is geometrically feasible just up to
$\sim$10~\% Co, which is the ``basic'' composition.  Thus, all TM-TM
neighbors must be Ni-Ni. Every Ni has mostly Al neighbors but cannot
escape having two or three Ni neighbors, since $\sim$30~\% of all
atoms are transition metals.

Let us check which ideal-site separations are favorable for which atom
pairs.  Within the same layer, the tile edge length $r_2$=2.45~\AA~ is
unfavorable for Al-Al or TM-TM bonds, but highly favorable for Al-TM
bonds. However, because of the high density of Al atoms, we find a
small number of Al-Al bonds do take this length. The short diagonal of
a fat rhombus is $r_3=2.88$~\AA, which is an acceptable Al-Al
distance. Hence the $72^{\circ}$ Al-TM-Al isosceles triangle (half a fat
rhombus) is highly favored within a layer.

The interlayer spacing $c/2$=2.04~\AA~ is too short for any pair.
Sites in adjacent layers, spaced by $\tau^{-1} a_0$ in-plane (e.g.,
the short diagonal of the thin rhombus), are separated by $r_1'=2.54$~\AA~
which is favorable for Al-TM or TM-TM bonds.  Finally, sites in
adjacent layers separated by $a_0$ in-plane have a total separation of
$r_2'=3.19$~\AA~ which is an acceptable Al-Al distance.

Given this understanding of chemical bond lengths we can easily
justify the decoration of the HBS tiles. Each tile is bounded by Al
atoms of alternating heights at separation $r_2'$. Interior sites of
the hexagon tile are too close to the Al border for Al atoms. Since it
must hold two TM atoms, it holds a pair of Ni atoms at distances
$r_1'$ and $r_2$ from the border Al and mutual separation $r_1'$.
Four of the border Al atoms form a rectangle with edges $r_2'$ and $r_3$
lying in a plane that is nearly the perpendicular bisector of the
Ni-Ni bond. This fragment of the hexagon tile is thus a slightly
distorted region of B2 (CsCl) structure~\cite{Steurer}.

The interior vertex of the boat and star tiles are at the ideal TM
distance $r_2$ from border Al atoms. Since this is a point of high Al
coordination, it is occupied by Co.  The boat and star tiles have room
for two additional interior Al atoms at separation $r_3$.  In an
isolated star tile this interior Al pair can lie in any of five
symmetry-related configurations. The structure surrounding the star
generally breaks this degeneracy by means of long-range interactions.

In the decoration just described, an Al atom on an HBS tiling vertex
is often at the center of a small cluster which was an important motif
of earlier models~\cite{Burkov,CW} and is confirmed by X-ray structure
refinement~\cite{Takakura}. This cluster appears in Fig.~\ref{small}b
wherever hexagons join at their tips.  This cluster consists of a
pentagon of mixed Al and Ni atoms in the same layer as the vertex Al,
and additional pentagons in the adjacent layers above and below. These
adjacent pentagons contain only Al atoms and are rotated by
$36^{\circ}$ with respect to the middle pentagon.  This cluster
exhibits interlayer Al-Ni separations of 2.54~\AA~ and 4.46~\AA,
precisely at the first and second minima of V${\rm AlNi}$.

Since the HBS tile corners are all multiples of $72^\circ$, edges
emanating from the HBS corner atoms in one layer can only point in the
five directions $+{\bf e}_i$ while those within the other layer point
in the directions $-{\bf e}_i$.  Statistically, the layers are
equivalent but related by a screw axis.  Allowing for the reflection
planes normal to the layers, and in the absence of further
symmetry-breaking, the HBS decoration implies a space group
$10_5$/mmc, consistent with experiment.

\subsection{Highly constrained simulation}
\label{constrained}

For the next level of modeling, we take the highly constrained HBS
tiles as fundamental objects.  Tile-tile interactions are defined
implicitly as the sum of the pair potentials between atoms decorating
the tiles.  The allowed ``flips'' (Monte Carlo moves) of the HBS
tiling are called ``bow tie flips'' as the tile edges before and after
the flip outline a bow tie shape~\cite{CW}.  The bow tie flips are
generated by fat hexagon flips of the underlying rhombus tiling.
Additionally the Al pair inside the star can rotate among its five
allowed orientations.  The reduced degrees of freedom make the highly
constrained HBS tiling much faster to simulate at low temperatures
than the minimally constrained rhombus tiling.

The ensemble of random HBS tilings contains a variable tile frequency
ratio H:B:S because HS pairs interchange with BB pairs by bow tie
flips. Highly constrained simulations forbid this flip because it
alters the chemical composition, given our ideal tile decoration.
There is a particular ``golden'' ratio H:B:S=$\sqrt{5}\tau:\sqrt{5}:1$
that is obtained, for example, by removing double-arrow edges from a
Penrose tiling. Decorating such a tiling deterministically yields an
ideal composition Al$_{0.700}$Ni$_{0.207}$Co$_{0.093}$ and atomic
volume 14.16~\AA$^3$. The composition is consistent with experiment,
while the atomic volume is low by about
5\%~\cite{Takakura,Cervellino}. The real structure probably
incorporates partial occupancy and is less strictly decorated than
this model.

\subsection{Supertiling}
\label{Supertiling}

Large-scale simulations (see Fig.~\ref{large}) reveal a ``supertile''
ordering in which hexagons connect tip-to-tip.  Each hexagon tip
becomes a vertex of a ``supertiling'', with longer edges of length
$\tau^2 a_0$ along the midline of every hexagon.  Since orientations
of adjoining hexagons differ by multiples of $72^\circ$ degrees, the
same is true for their midlines, hence the ``supertile'' edges differ
by $72^\circ$ angles and form mainly HBS tiles (as well as a new
``defect'' tile discussed below). Each HBS supertile has Al atoms at its
vertices, Ni atom pairs along its edges, and Al and Co atoms at
special points in its interior. The only variability in the supertile
decoration occurs in the asymmetric placement of one (in H and B) or
two (in S) interior Al atoms.

The supertile atomic structure is mechanically stable.  Under relaxation of
the structure shown in figure ~\ref{small} (which consists of two large
scale hexagons), the average Co and Ni displacement is just
0.10~\AA. The average Al displacement is 0.17~\AA~ except for the two
Al atoms located slightly off-center in the large-scale hexagons which
displace 1.13~\AA~ to the symmetric points at the hexagon centers.

The energetics of the HBS supertilings follows a ``tile Hamiltonian''
in which we replace the actual interatomic interactions with effective
interactions within and between the tiles. The tiles are decorated
deterministically as in Fig.~\ref{large}a. The functional form
\begin{equation}
\label{TH}
H=c_0 N_{atoms} + c_{72} N_{72} + c_{144} N_{144}
\end{equation}
provides a reasonably simple and accurate description. Here
$N_{\theta}$ is the number of vertices of angle $\theta$.  We obtained
energy data from a Monte Carlo simulation at $k_B T=0.1 eV$ during
which we monitored the energies of 1400 accepted flips of an $L_x
\times L_y \times 4$~\AA$^3$ approximant structure containing 1466
atoms in its unit cell. The resulting coefficients are $c_0=-0.246
eV$, $c_{72}=0.218 eV$, $c_{144}=-0.056 eV$, and the r.m.s.  deviation
of the actual energies from this fit is 0.021 eV. Of this deviation,
we account for 0.007 eV simply from the placement of symmetry-breaking
Al atoms in the supertile interiors.

Note that 72$^{\circ}$ angles are disfavored ($c_{72}>0$) because they
increase the number of NiNi neighbors and thereby reduce the number of
favorable Al-Ni neighbors. Flips of the supertiling that replace an HS
pair with a BB pair result in a net reduction of one 72$^{\circ}$
angle. Thus low energy structures of the HBS supertiling contain few
or no S tiles. Our Hamiltonian depends only on the numbers of tiles of
each type and contains no interactions between tiles. Consequently it
has a highly degenerate ground state. A more accurate Hamiltonian
would contain some weak terms that couple to the arrangement of the
tiles.  Additionally, we do not include stacking disorder. A large
term should be added to eq.~(\ref{TH}) to model the energy cost of a
mismatch in the tiling between successive 4~\AA~ slabs.

The ``defect'' tile, with a shape we call a ``bow tie'', breaks the
connectivity of the small-hexagon chain.  Defect tiles enter only when
they reduce the number of 72$^{\circ}$ junctions.  They accomplish
this reduction by interchanging a Ni-Ni pair with a nearby Al-Co pair.
We can include the defect in our tile hamiltonian by adding an
additional term $c_{defect} N_{defect}$ with $c_{defect}=0.020 eV$.
Despite the positive sign of this term, defects lower the total energy
by virtue of a reduction in $N_{72}$.

An alternate means of reducing the frequency of 72$^{\circ}$ angles
between tile edges decorated with Ni atom pairs is to alter the
chemical composition. To accommodate the new composition we must relax
certain constraints in our simulation.  We keep the small scale HBS
tiles with Al fixed on their boundaries, but allow arbitrary chemical
occupancy of the interior sites.  Replacing 20~\% of NiNi pairs with
AlCo pairs eliminates all bow tie tiles from the minimum energy
configuration and leads to chemical composition
Al$_{0.720}$Ni$_{0.166}$Co$_{0.114}$, still within the limits of the
Basic Ni composition. The low energy configurations consist entirely
of HBS tiles decorated as found previously, but now many tile edges
that participate in two 72$^{\circ}$ junctions get decorated with an
AlCo pair rather than a NiNi pair (see Fig.~\ref{large}b).

\section{Comparison}
\label{Comparison}

We compare our model with structural models based on several recent
experiments.

Z-contrast electron microscope imagery~\cite{Z-contrast} images atomic
columns proportionally to the mean square atomic number of the column,
so the images translate quite directly into TM positions. A key
feature of the experimental data is the occurrence of decagonal rings
with a 20~\AA~ diameter. These rings exhibit up to 10 TM doublets
around the perimeter, an interior ring of 10 TM singlets, and a
central triangular core~\cite{Z-contrast}. This characteristic
structure is in excellent agreement with our model, where two hexagons
and a boat frequently coalesce into a decagonal cluster (see center of
Fig.~\ref{large}b). The triangular core of this cluster, which breaks
decagonal symmetry, is recognized as the sail of the boat tile. Full
{\it ab-initio} calculations~\cite{Yan} recently verified energetic
favorability of this particular core structure~\cite{Caveat}.

One disagreement between our work and the Z-contrast imagery is we
find a lower apparent density of complete 20~\AA~ decagonal rings. It
is conceivable that phason stacking disorder (tilings differ in
adjoining 4~\AA~ slabs by localized bow tie flips) increases the
apparent density of decagonal rings~\cite{CLH}. Alternatively, partial TM
occupancy may be responsible as discussed in connection with our
defect tile.

We can also compare our results with recent structure refinements
based on X-ray data sets. The refinements by Takakura, et
al.~\cite{Takakura} and by Cervellino, et al.~\cite{Cervellino}
resemble our model in several key respects. In particular, both
feature frequent TM doublets along HBS tile edges. Polarized EXAFS
experiments~\cite{Zaharko} confirm that Co atoms are isolated from
other TM atoms, and Ni atoms do have Ni neighbors at positions
consistent with our edge decoration model.  Sites where the Al atoms
of AlCo pairs replace NiNi pairs in our model are among the sites
where X-ray data indicates partial TM occupancy (specifically Al/TM
mixing on orbits 6 and 7 of ~\cite{Takakura}, partial Ni occupancy of
~\cite{Cervellino}).

In fact, the partial TM occupancy or AlTM mixing is more prevalent in
the structure refinements~\cite{Takakura,Cervellino} than in our
model, even at the reduced Ni composition. This may be a finite
temperature effect in the real materials, while our modeling effort
discussed here concerns minimum energy states. Indeed, we do find AlTM
mixing at elevated temperatures. Additionally, increasing the atomic
volume by reducing the total density will cause significant partial
occupancy.

Another small difference between our model and the experimental
refinements is the asymmetric placement of interior Al atoms in the S
and B supertiles. These are the two Al atoms closest to the central Co
in the small 2.45~\AA~ edge length S tiles that are contained within
the S and B supertiles. Takakura et al.~\cite{Takakura} find the
occupied sites lie at positions (their orbit 23) rotated by
36$^{\circ}$ from our positions (their orbit 10), while
Cervellino et al.~\cite{Cervellino} find them at both types of
positions, within the ``puckered'' layers. In our simulations the
alternate positions are energetically unfavorable, although they
become favorable after small relaxations of the atoms at those
positions.

Our study began with interatomic potentials plus a minimum of
experimental information. We derived structural models starting with a
minimally constrained lattice gas on a fluctuating rhombus tiling.
Systematically removing unnecessary degrees of freedom yielded a
nearly deterministic decoration of HBS tiles at a length scale
$\tau^2$ times larger than the initial rhombus tiling, a model
consistent with Z-contrast electron microscopy and X-ray diffraction.
This procedure can be repeated to identify yet larger characteristic
atomic clusters providing a novel example of multiscale modeling which
might be applicable to other quasicrystals.

\thanks{This research was supported in part by the National Science Foundation 
grant DMR-0111198 at CMU and by DOE grant no. DE-FG02-89ER-45405 at
Cornell. Part of this work was performed under the auspices of the US
Department of Energy by Lawrence Livermore Laboratory under contract
W-7405-ENG-48. We acknowledge useful discussions with W. Steurer, A.
Cervellino, H.  Takakura, E. Abe, P.J. Steinhardt and S.J. Pennycook}



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\newpage

\begin{table}[!ht]
\begin{tabular}{|l|l|l|}
\hline
\hline
Label           & Example                              & Comments \\
\hline
$r_1$=1.51~\AA  & $a_0({\bf e}_0+{\bf e}_3$)           & Forbidden   \\
$r_2$=2.45~\AA  & $a_0 {\bf e}_0$                      & Al-TM, (Al-Al) \\
$r_1'$=2.54~\AA & $a_0({\bf e}_0+{\bf e}_3)+{{c}\over{2}}\hat{z}$ 
                                                       & Al-TM, TM-TM  \\
$r_3$=2.88~\AA  & $a_0({\bf e}_1-{\bf e}_0)$           & Al-Al \\
$r_2'$=3.19~\AA & $a_0 {\bf e}_0+{{c}\over{2}}\hat{z}$ & Al-Al \\
\hline
\hline
\end{tabular}
\caption{Characteristic distances (illustrated in Fig.~\ref{layer}) 
and important bond types. Primed vectors connect adjacent layers.}
\label{tab1}
\end{table}

\newpage

\begin{figure}
\vspace{-0.6 in}
\centerline{\epsfig{file=fig1.eps,width=6in}}
\vspace{2 in}
\caption{(a) GPT pair potentials for Al-Al, Al-Co and Al-Ni. 
(b) Modified GPT pair potentials for Co-Co, co-Ni and Ni-Ni.}
\label{pot}
\end{figure}

\newpage

\begin{figure}
\vspace{-0.6 in}
\centerline{\epsfig{file=fig2.eps}}
\vspace{-1 in}
\caption{Random rhombus tiling decorated with ideal sites. The
long- and short-dashed lines outline, respectively, thin and fat
hexagons.  Unit vectors $\{{\bf e}_i\}$ lie parallel to tile
edges. Vectors ${\bf r}_i$ are defined in the table.}
\label{layer}
\end{figure}

\newpage

\begin{figure}[!ht]
\centerline{
\epsfig{file=fig3a.eps,width=4in}}
%\vspace{-5pt}
\centerline{
\epsfig{file=fig3b.eps,width=4in}}
\caption{Minimum energy configurations. Small/large circles indicate atoms in
upper/lower layer. Gray=Al, Blue=Co, Black=Ni, White=vacant. (a) Top
figure results from minimally constrained simulation (solid/dashed
lines denote upper/lower tilings). (b) Bottom figure results from highly
constrained simulation. Dark solid lines outline $a_0$-scale HBS
tiling.  Pink-shaded hexagons connect vertices of $\tau^2 a_0$-scale HBS
tiling.}
\label{small}
\end{figure}

\newpage

\begin{figure}[!ht]
\centerline{
\epsfig{file=fig4a.eps,width=4in}}
%\vspace{-15pt}
\centerline{
\epsfig{file=fig4b.eps,width=4in}}
\caption{Lowest energy configurations obtained.
(a) Top shows highly constrained simulation. Green lines outline bow
tie tiles. (b) Bottom variable occupancy simulation. Small hexagon
tiles, occupied by either NiNi pairs or AlCo pairs, lie on edges of an
HBS supertiling in (b).}
\label{large}
\end{figure}

\enddocument

