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\newcommand{\alco}[2]{\mbox{Al$_{#1}$Co$_{#2}$}}
\newcommand{\alcu}[2]{\mbox{Al$_{#1}$Cu$_{#2}$}}
\newcommand{\alfe}[2]{\mbox{Al$_{#1}$Fe$_{#2}$}}
\newcommand{\almn}[2]{\mbox{Al$_{#1}$Mn$_{#2}$}}
\newcommand{\alni}[2]{\mbox{Al$_{#1}$Ni$_{#2}$}}
\newcommand{\alos}[2]{\mbox{Al$_{#1}$Os$_{#2}$}}
\newcommand{\lsb}{\mbox{L1$_2$}}
\newcommand{\dosb}[1]{\mbox{D0$_{#1}$}}
\newcommand{\alconi}[3]{\mbox{Al$_{#1}$Co$_{#2}$Ni$_{#3}$}}
\newcommand{\alcocu}[3]{\mbox{Al$_{#1}$Co$_{#2}$Cu$_{#3}$}}
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\title{First-principles interatomic potentials for transition-metal
aluminides.  III. Extension to ternary phase diagrams}

\author{Mike Widom and Ibrahim Al-Lehyani}

\address{Department of Physics, Carnegie Mellon University, Pittsburgh,
Pennsylvania 15213}

\author{John A. Moriarty}

\address{Lawrence Livermore National Laboratory, University of
California, Livermore, California 94551}

\date{\today}
\maketitle

\begin{abstract}
Modeling structure and mechanical properties of intermetallic alloy
requires detailed knowledge of their interatomic interactions. The
first two papers of this series [Phys. Rev. B {\bf 56}, 7905 (1997);
{\bf 58}, 8967 (1998)] derived first-principles interatomic potentials
for alloys using generalized pseudopotential theory (GPT). Those
papers focused on binary alloys of aluminum with first row transition
metals and assessed the ability of GPT potentials to reproduce and
elucidate the alloy phase diagrams of Al-Co and Al-Ni. This paper
addresses the binary alloy Al-Cu and the ternary compounds Al-Co-Cu
and Al-Co-Ni using pair potentials calculated in the limit of
vanishing transition metal concentration. Despite this approximation,
we find tolerable agreement with the known T=0K phase diagrams, up to
50~\% total TM concentration provided the Co fraction is below 25~\%.
Composition-dependent potentials and many-body interactions would be
required to correct deficiencies at higher Co concentration. Outside
this troublesome region, the experimentally determined stable and
metastable phases all lie on or near the convex hull of a scatter plot
of energy versus composition. We verify, qualitatively, reported
solubility ranges extending binary alloys into the ternary diagram in
both Al-Co-Cu and Al-Co-Ni. Finally, we reproduce previously
conjectured transition metal positions in the decagonal quasicrystal
phase.
\end{abstract}

\pacs{71.15.Nc,71.15.Mb,61.66Dk,61.43.Bn}


\section{Introduction}
\label{sec:intro}

Intermetallic alloys exhibit scientifically interesting and
technologically important structures and properties.  Their
special mechanical, thermal and chemical properties invite
practical applications~\cite{Westbrook,Schwartz}. Their
complicated and fascinating crystal~\cite{Shoemaker} and
quasicrystalline~\cite{QCReviews} structures are of fundamental
scientific interest. Because atomic interactions and cohesion
govern crystal structure and important mechanical properties such
as elastic constants of solids, understanding the interactions
among atoms within these compounds should deepen our understanding
of their novel structures and properties\cite{Hafner,Pettifor}.

Investigating the interatomic interactions in intermetallic compounds
is a considerable theoretical challenge. Ab-initio
methods~\cite{AbInitio} become computationally demanding, due to the
large unit cell with many inequivalent atomic sites and the likelihood
of structural and chemical disorder. Interatomic potentials~
\cite{Moriarty8283,Moriarty88,Krajci92,Phillips94,Windisch94pot,Mihalk}
are well suited to complex or disordered structures. However, the
multiplicity of chemical species requires the calculation of
numerous interaction potentials, and the presence of transition
metals may require that angular dependent many-body interactions
be considered. Further, the dissimilar electronic structure of
simple metals compared with transition metals requires a mixed
basis for electronic states containing plane waves for $sp$
electrons and tight binding orbitals for transition metal $d$
electrons.

Previous papers (henceforth referred to as paper I~\cite{paper_I}
and paper II~\cite{paper_II}) introduced the formalism of the
generalized pseudopotential theory (GPT) for treatment of
aluminum-rich binary intermetallics, and studied their
applicability to the compounds \alco{}{} and \alni{}{} in detail.
This paper extends that approach to interatomic potentials for
ternary intermetallic alloys. Previously calculated ternary alloy
interactions~\cite{Krajci92,Windisch94pot} treat simple metals
with empty d-shells and noble metals with completely filled
d-shells. The present paper is thus the first treatment of
interatomic interaction potentials in ternary alloys containing
transition metals.

Our goal in the present paper is to test the ability of the GPT
interatomic pair potentials to reproduce the experimentally known
ternary phase diagrams. Thus we examine the mechanical stability of
known structures against atomic displacements, and we examine the
thermodynamic stability against decomposition of an alloy into phases
of differing composition. Requiring mechanical and thermodynamic
stability of all known phases, and at least thermodynamic instability
of a set of hypothetical structures, places stringent constraints on
the atomic interactions.

Our work focuses on the compound Al-Co-Cu with some brief
comparisons made with Al-Co-Ni. There are several reasons
for this choice. A primary motivating factor is the existence of
stable decagonal quasicrystal phases in these compounds with
reasonably well understood atomic structure. Furthermore, Al-Co-Cu in
particular has numerous stable phases in the ternary diagram (see
Fig.~\ref{fig:ternaries}a) with well defined compositions that are
well isolated in composition and distinct in structure from binary
alloys. These ternary phases appear at compositions where we may
expect our potentials to apply with reasonable accuracy. Studying
Al-Co-Cu gives us the chance to extend to the Al-Cu binary the degree
of attention already devoted to Al-Co and Al-Ni. Finally, there will
be interesting comparisons to make between the Al-Co-Cu and Al-Co-Ni
potentials and phase diagrams. In Al-Co-Ni the composition fields of
structures exhibit strong miscibility between Co and Ni (see
Fig.~\ref{fig:ternaries}b) in contrast to the behavior of Al-Co-Cu.

To test the ability of our potentials to reproduce the experimentally
observed phase diagram (at T=0K) we must demonstrate that the known
low temperature structures define the vertices of the convex hull of a
scatter plot of cohesive energy versus composition. All other
conceivable structures must lie above this convex hull. We cannot
examine all conceivable structures, so we restrict our attention to
structures that are either widely occurring simple structures or more
complex structures observed in chemically similar compounds.

We find that our ternary potentials apply tolerably well within
most regions of the \alcocu{1-x-y}{x}{y} phase diagram of greatest
interest. Along the the binary \alcu{1-y}{y} axis we find nearly
perfect agreement with the known stable and metastable phase
diagram up to $y=1/2$ using only the potentials evaluated at
$y=0$. The only failure is the spurious appearance of the
$\tau$ structures (described below in section~\ref{sec:binary})
on the convex hull. Extending into the ternary
\alcocu{1-x-y}{x}{y} phase diagram, we find tolerable agreement
provided $x+y<1/2$ and $x<1/4$. In this regime the only clear
disagreement between our results and the known phase diagram is that
the known stable structure \alcocu{7}{}{2} lies slightly (9 meV/atom)
above the convex hull in our calculation. For $x~\ge~1/4$ our ternary
calculations fail because of known inadequacies of the binary
\alco{1-x}{x} calculation~\cite{paper_II}.  Specifically, we do not
adequately address the vacancy concentration in the O-\alco{13}{4}
structure and we cannot treat \alco{5}{2} at the pair potential level.
Encouragingly, the decagonal quasicrystal phase lies in a region of
the phase diagram where our potentials may be expected to apply
reasonably well.

The following section reviews the formalism and limitations of the
generalized pseudopotential theory as applied in this paper.
Section~\ref{sec:binary} applies the GPT potentials to calculate
the binary alloys phase diagram of Al-Cu, followed by a treatment
of ternary phase diagrams in section~\ref{sec:ternary}. In
section~\ref{sec:QC} we discuss the utility of these potentials
applied to decagonal quasicrystal structures.

\section{Total energy calculations}
\label{sec:totE}

Paper I described the theoretical basis for calculating
interatomic potentials within the generalized pseudopotential
theory. Here, we review some key ideas.  The GPT interatomic
potentials are terms in a real-space expansion of total energy in
the form of a volume energy, pair and many-body interactions:
\begin{eqnarray}
\label {eq:etot}
E_{\rm tot}({\bf R}_1...{\bf R}_N) & = & NE_{\rm vol}(\Omega ,{\bf c}) +
{1\over 2} \sum _{\alpha ,\beta} \sum _{i,j}
\mbox{}^\prime \ v^{\alpha \beta}_2(R_{ij};\Omega ,{\bf c})
\nonumber \\
& & + {1\over 6} \sum _{\alpha ,\beta, \gamma} \sum _{i,j,k}
\mbox{}^\prime \ v^{\alpha \beta \gamma}_3
(R_{ij},R_{jk},R_{ki};\Omega ,{\bf c})
\nonumber \\
& & + {1\over 24} \sum _{\alpha ,\beta, \gamma, \delta} \sum _{i,j,k,l}
\mbox{}^\prime \ v^{\alpha \beta \gamma \delta}_4
(R_{ij},R_{jk},R_{kl},R_{ik},R_{jl},R_{il};\Omega ,{\bf c})
+ \cdots \ .
\end{eqnarray}
Here ${\bf R}_1...{\bf R}_N$ denotes the positions of the $N$
ionic cores and the prime on each sum over ion positions excludes
all self-interaction terms. $\Omega$ is the atomic volume and {\bf
c} is a vector whose elements depend upon the concentration of
different chemical species. Indices $\alpha, \beta, \gamma,
\delta, \cdot\cdot\cdot$ run over all chemical species, and indices
$i, j, k, l, \cdot\cdot\cdot$ run over the individual ions $1,
\cdot\cdot\cdot, N$.

The volume term $E_{\rm vol}(\Omega,{\bf c})$ is structure
independent. It exerts no force on the individual atoms, but is
important for determining the cohesive energy, equilibrium volume, and
bulk modulus.  The sums over the pair-potentials ($v_2$) are the
leading structure-dependent terms in the total energy.  The many-body
interactions ($v_3$ and $v_4$) are presumed to be strongest among
clusters of transition-metal atoms, due to the influence of their $d$
electrons, and weaker among clusters containing aluminum atoms.
Consequently, the many-body interactions should be negligible at low
transition-metal concentration, and grow progressively more important
at higher transition-metal concentration.  In pure elemental
transition metals, the three- and four-body interactions are
important, but higher-order interactions may often be
neglected~\cite{MeltingMo}.  In general, both the pair and many-body
potentials are long-ranged with oscillatory tails arising from
electron screening and/or $sp$-$d$ hybridization. One can often
demonstrate relationships between the oscillations of the potentials
and favored or disfavored crystal
structures~\cite{Hafner,Pettifor,paper_I,paper_II,WC96}.

All terms in the GPT total-energy expansion depend on the atomic
volume and chemical composition.  The discussion in the remainder
of the paper uses only the GPT potentials evaluated in the limit
of vanishing transition metal concentration, motivated by the
observation~\cite{Phillips94} that the valence electron density
varies slowly with $x$, near $x=0$, for \alco{1-x}{x} compounds.
We confirm in paper II that the limiting potentials achieve
considerable success, but find that certain details in the alloy
phase diagrams requires the composition-dependent GPT.

Calculated pair potentials for the Al-Co-Cu and Al-Co-Ni systems,
in the aluminum-rich limit,
are displayed in Fig.~\ref{fig:potentials}.
Figure~\ref{fig:potentials}a shows interactions of Al atoms with
themselves and with Cu, Ni and Co. It is noteworthy that Al
near-neighbor interactions (below about 3~\AA) are strongly
disfavored compared to interactions with transition metals such as
Ni and Co. At low TM concentration this tends to favor structures
with widely spaced transition metal atoms so as to maximize the
number of Al-TM near neighbor bonds.

Figure~\ref{fig:potentials}b displays the self-interactions of the
same transition metal atoms. We actually do not calculate the mixed
transition metal potentials $v^{\alpha\beta}_2$ with $\alpha \ne
\beta$ explicitly.  Rather, we make approximations based upon first
order expansions of the total energy in the atomic number difference
$Z_{\alpha}-Z_{\beta}$.  Thus the $v^{\rm CoNi}_2$ potential is set to
the average $(v^{\rm CoCo}_2+v^{\rm NiNi}_2)/2$, and for $v^{\rm
CoCu}_2$ we simply employ $v^{\rm NiNi}_2$.  Plausibility of these
approximations is supported by noting how close the Ni-Ni potential
lies to the average of the Co-Co and Cu-Cu potentials.  One final
point to note in Fig.~\ref{fig:potentials}b is the strong binding of
Co-Co pairs at unphysically short distances.  This feature is a known
difficulty of the pair interactions for TM near neighbors.  In reality
the Co atoms repel at these distances due to contributions that enter
the total energy only at the three- and four-body potential level in
our expansion.

While we find tolerable qualitative agreement between our
calculations and the experimentally determined phase diagrams, we
do encounter certain difficulties that we believe can be traced
back to the approximations employed. First of all, taking the
limit of vanishing transition metal concentration, and making the
assumption of constant electron density, eliminates the variation
of volume energies and pair potentials with composition and atomic
volume. This causes small systematic errors in our calculated
total energies and diminishes our ability to address vacancies and
substitutional disorder over a composition range. Secondly, by
dropping the many-body interactions we introduce significant
errors in the energies of structures containing transition metal
near neighbors. We lose important angle-dependent effects, and we
encounter difficulties with unphysically strong transition metal
attractions.

These difficulties are illustrated in our findings reported in paper
II.  There we found that we could reproduce the known phase diagram
for \alco{1-x}{x} up to $x=0.264$ using only potentials calculated in
the $x=0$ limit. However, the orthorhombic and monoclinic variants of
\alco{13}{4} appeared with substantially higher vacancy concentrations
than the latest experimental assessments~\cite{Grin,Freiburg} place
them. Consideration of composition and atomic volume dependent
potentials should resolve these difficulties.  Furthermore, at
$x=0.2857$, the pair potentials favored
\alfe{5}{2} over the true structure of \alco{5}{2}. Inclusion of
3- and 4-body interactions resolved this problem. In the case of
Al-Ni we found, using the $x=0$ potentials, that the \alco{9}{2}
structure incorrectly falls on the convex hull, and that
\alco{13}{4} preempts the \dosb{11} structure of \alni{3}{}. Both
these difficulties were alleviated by use of the composition and
atomic volume dependent potentials.

\section{Al-Cu binary phase diagram}
\label{sec:binary}

Since Cu is a noble metal with a completely filled d-shell its
many-body interactions should be relatively weak compared with
interactions among transition metal atoms. That suggests that the GPT
could apply to the compound \alcu{1-y}{y} for all $y$ from 0 to 1,
keeping only the volume energy and the 2-body interactions. For our
present purposes, however, we will focus on the Al-rich side, up to
$y=1/2$, because this is the concentration range that interests us at
present and because we employ potentials calculated in the limit
of vanishing Cu concentration.

The aluminum-rich side of the \alcu{1-y}{y} phase
diagram~\cite{JLMurray} is very sparse.  Between pure FCC aluminum
at $y=0$ and \alcu{}{} at $y=1/2$ there exists only one stable
phase, \alcu{2}{} at $y=1/3$. In contrast, the copper-rich side
contains a large number of phases, many with both low- and
high-temperature variants, and many with complex or unknown
structure. Even \alcu{}{}, at $y=1/2$, has both a complicated
low-temperature (LT) structure~\cite{Boragy} mC20 and a high
temperature (HT) variant, either oP16 or oC16 of unspecified
structure~\cite{Preston}. Both variants of \alcu{}{} are
considered to be vacancy-ordered phases based upon the B2 (CsCl)
structure.

Table~\ref{tab:alcu} lists the structures considered in our
evaluation of the Al-Cu binary potentials. Most of these
structures are self-explanatory. For example, Cu replaces the
transition metal in Al-TM compounds. In other cases the assignment
of Al and Cu atoms among the sites may be deduced from the overall
stoichiometry. Certain structures require specific comment because
of choices made about their composition or because their
structures are unfamiliar. The orthorhombic phase O-\alco{13}{4}
is taken to be fully occupied~\cite{Grin}. We incorporate vacancies in
\alconi{75}{22}{3} by removing one Al from each pair with separation
less than 2.4~\AA~as discussed in paper II. The monoclinic phase
M-\alco{13}{4} incorporates vacancies as proposed by Hudd and
Taylor~\cite{Hudd}. However, its extension to the ternary
M-(Al,Cu)$_{13}$Co$_4$ utilizes the fully occupied structure~\cite{Freiburg}.

An interesting family of vacancy-ordered phases are the so-called
$\tau$ phases~\cite{tau} that occur as metastable phases in binary
Al-Cu~\cite{Sastry} and as metastable or stable phases in
Al-Ni~\cite{Chattopadhyay} and in ternary systems such as
Al-Cu-Ni~\cite{tau} and Al-Co-Cu~\cite{He88}. These structures are
based upon the B2 structure, with compositional modulation along
the [111] direction. In the B2 structure of Al-TM alloys the (111)
planes are occupied, alternately, by Al or by TM atoms. The
compositional modulation deletes certain TM planes, with the
pattern of occupied (O) or vacant (V) TM planes following a
``Fibonacci'' sequence\cite{Chattopadhyay}. For example, in the
$\tau_3$ structure every $3^{\rm rd}$ TM layer is vacant in the
pattern OVO. This is the structure of the stable phase \alni{3}{2}
and the metastable phase \alcu{3}{2}. The $\tau_5$ structure is
the pattern OVOVO, $\tau_8$ is formed by joining a single repeat
unit of $\tau_3$ to a single repeat unit of $\tau_5$, and
$\tau_{13}$ joins $\tau_5$ with $\tau_8$.

Because the structure of the HT phase at composition $y=1/2$ is
unknown we created a test structure consistent with the known lattice
constants and symmetries. Thus we start with a B2 structure with
lattice constant $a_0$. The unit cell of our HT contains 18 unit cells
of the B2 structure and has lattice constants $(\sqrt{2},3\sqrt{2},3)$
in units of $a_0$. We introduce a pair of Al vacancies into one layer
perpendicular to the c axis and a pair of Cu vacancies into a layer
c/2 away. The choice of vacant sites is made consistently with C-face
centering, producing an oC16 structure.

For each of the structures listed in table~\ref{tab:alcu} we
isotropically scaled the lattice constants so that the free electron
density matched that of pure FCC aluminum (0.18076 \AA$^{-3}$). We
employ a free electron valence of 3 for each Al atom and 1.805 for
each Cu atom~\cite{paper_I}. The structures were then relaxed until
the average force per atom dropped below 10$^{-3}$ eV/\AA, then their
total energy is reported. Figure~\ref{fig:Eofy} shows a scatter plot
of the data. We classify the stability of structures according to
their displacements upon relaxation and display this information using
plotting symbols. Highly symmetric structures do not relax because
forces balance exactly. Certain structures had displacements beyond
0.3~\AA/atom. We consider those structures to be unstable and display
the unrelaxed energies of these structures.  To remove a strong
linear slope of the energy vs. composition data, we subtract all
energies from the tie-line joining FCC aluminum to
\alcu{3}{} in the \lsb~ structure, defining $\Delta E$.

Inspection of fig.~\ref{fig:Eofy} reveals that the convex hull is well
respected.  Up to $y=0.5$ there is only a single unreported structure,
the $\tau_{5}$ structure at $y=0.375$, that touches the convex hull.
The metastable $\tau_3$ structure of \alcu{3}{2}, and the high
temperature variant of \alcu{}{} both lie slightly above the convex
hull, consistent with their near stability.  Similarly,
Al$_7$Cu$_4$Li, with an Al atom replacing Li, nearly touches the
convex hull. Extending into the AlCuLi ternary diagram could possibly
stabilize this structure. Both \alco{9}{2} and \dosb{11} (the
structure of \alni{3}{}) lie above, but close to, the convex hull as
is appropriate considering the proximity of Co and Ni to Cu in the
periodic table.

\section{Ternary phase diagrams}
\label{sec:ternary}

Ternary phase diagrams for both \alconi{}{}{} and \alcocu{}{}{}
are known in some detail~\cite{Godecke,Gru93}. The labeled regions in
Fig.~\ref{fig:ternaries} represent the known stable phases.
\alcocu{}{}{} (Fig.~\ref{fig:ternaries}a) exhibits several distinct
structures within well separated composition fields~\cite{Gru93}. The
ternary compounds include \alcocu{7}{}{2} (a tP40 structure containing
cubes of Al centered with Cu atoms~\cite{Bown}), the decagonal
phase~\cite{D-AlCoCu}, and a pair of phases ($\tau_3$ and $\tau'$)
believed to be related to the vacancy stabilized phases described
above in the context of binary \alcu{}{} structures. At 50\% Al
concentration, the B2 (CsCl) structure of Al(Cu,Co) extends from AlCo
nearly to AlCu. It is believed that vacancies proliferate within the
ternary B2 structure~\cite{Ridley}.  The actual AlCu structure, as
noted above, is a vacancy ordered B2 structure.

It is intriguing that the M-(Al,Cu)$_{13}$Co$_4$ phase (well known as
a decagonal quasicrystal approximant), the decagonal phase composition
field, the $\tau$ phases and the vacancy stabilized \alcu{}{} phase
are nearly co-linear in this phase diagram, lying near the line
$2x+y=1/2$ connecting \alco{3}{} with \alcu{}{}.  Along this line Cu
substitutes for both Al and Co in equal numbers. This effect motivated
construction of successful ``mock ternary'' AlCoCu
potentials~\cite{CW98} that started with binary AlCo
potentials~\cite{Phillips94} then defined the interactions of Cu atoms
as if Cu was a linear superposition of Al and Co.  Remarkably,
Raynor's~\cite{Raynor} effective valences of $+1$ for Cu and $-1$ for
Co, yield a constant electron per atom ratio of 2 along this line,
suggesting a possible link between this electron to atom ratio and
formation of the quasicrystal phase~\cite{eoveraratio}.

The phase diagram~\cite{Godecke} of \alconi{}{}{}
(Fig.~\ref{fig:ternaries}b) contrasts strongly~\cite{Gru94} with
\alcocu{}{}{}. Instead of well isolated composition fields, the phase
diagram of \alconi{}{}{} is marked by highly elongated composition
fields, with Ni and Co substituting for each other, but not replacing
Al. The miscibility of Co and Ni in the B2 structure is complete. Many
other binary structures extend far across the ternary diagram.

We begin our exploration of the ternary phase diagram with the
compound \alcocu{1-x-y}{x}{y}. Figure~\ref{fig:alcocu} reproduces the
known ternary diagram and superposes some information about our
calculations. Table~\ref{tab:alcocu} lists ternary structures
representative of various points in the composition plane.  The
labeled circles represent the experimentally known structures we
examined. The black triangles locate compositions at which we tested
hypothetical structures. The dashed tie-lines define the edges of our
calculated convex hull.

In some cases, where a real or hypothetical structure is not fully
known, we made reasonable guesses. For example, we employed the
Cockayne and Widom~\cite{CW98} model for the decagonal phase. A
quasicrystal has no unit cell, so we actually studied a crystalline
approximant with unit cell dimensions $60\times 51\times 4$~\AA$^3$ of
composition \alcocu{586}{178}{142}. The fully occupied
M-(Al,Cu)$_{13}$Co$_4$ structure is taken from Freiburg and
Grushko~\cite{FG} in which Cu positions are specified. The ternary
$\tau_3$ phase is based upon the $\tau_3$ structure of \alcu{3}{2},
with the unit cell doubled along the 3-fold axis. A single Cu atom is
then replaced with a Co atom to create \alcocu{6}{}{3}.  The ternary
$\tau'$ structure is based upon the $\tau_{13}$ unit cell, taking
composition \alcocu{36}{3}{24} close to the experimentally reported
composition. A simulated annealing procedure established the optimal
sequence of Al, Co and Cu layers holding the atomic sites fixed at
their ideal positions. Near-neighbor Co atoms were prohibited during
the annealing in order to avoid overbinding at short distances.

A pictorial summary of our quantitative results is achieved by
plotting total energies along special lines in the diagram. The
special lines we choose are (a) $x = y$ (see Fig.~\ref{fig:Eofx=y}),
(b) $x+y = 1/4$ (see Fig.~\ref{fig:Eofx+y=1/4}) (c) $x+y = 1/3$ (see
Fig.~\ref{fig:Eofx+y=1/3}) because many simple and well known
structures lie on these lines.  Data point symbols are as in the
corresponding figure for binary \alcu{}{} except for the addition of
$+$ symbols marking the convex hull and $\times$ symbols marking the
intersections of our special lines with tie-lines between the two
structures named adjacent to the marker. When one of these tie-lines
meets the convex hull, the $+$ and $\times$ symbols overlay each other.

For the most part, our calculated energies are consistent with the
experimentally known phase diagram. The hypothetical structures lie
above the convex hull, so that only known structures (or tie-lines
joining them) lie on the hull. We now focus on a few points of
discrepancy between our data and the experimental phase diagram. These
points of disagreement indicate the limits of reliability of the GPT
energies truncated at the pair potential level and neglecting
composition and volume dependence.

An immediately apparent problem is that no edge (dashed line) of the
convex hull in Fig.~\ref{fig:alcocu} joins the experimentally known
\alcocu{7}{}{2} structure, and thus that structure does not occupy a
vertex of the convex hull. Indeed, the calculated energy of this
structure lies 0.009 eV/atom above the convex hull.  Likewise, no
convex hull edge joins \alco{5}{2}. This was expected from our prior
discussion and reflects the need to include many-body interactions.

These same difficulties are evident in our quantitative plots of
energy versus composition along special lines.  As noted above,
\alcocu{7}{}{2} lies slightly above the convex hull, and thus
tie-lines connecting to it lie slightly above the hull. See the line
$x=y$ in Fig.~\ref{fig:Eofx=y} for some examples. Along the lines
$x+y=1/4$ and $x+y=1/3$ we expect difficulty near $y=0$ due to the
known inadequacy of $x=0$ pair potentials for \alco{1-x}{x} at large
$x$. For reference, we have placed the calculated energies of some
hypothetical binary structures on the energy diagram for $x+y=1/4$ at
$y=0$. These fall below the convex hull of the energies for the structures
we considered when calculating the ternary compounds. At $y=0$ on the
line $x+y=1/3$ (see Fig.~\ref{fig:Eofx+y=1/3}) the difficulty is that
\alco{}{} coexists with O-\alco{13}{4} rather than with
\alco{5}{2}.

\subsection{Composition ranges}
\label{ranges}

The ability of potentials to reproduce a trend among distinct
compounds is a further test of applicability. We have examined two
special lines in the \alconi{}{}{} phase diagram and contrasted them
with the same lines in the \alcocu{}{}{} diagram. Our goal is to
understand the differing solubilities of one transition metal for the
other.

Along $x+y=0.1818$ we study the stability of the \alco{9}{2} structure
as Cu or Ni replaces Co. As soon as we replace one out of the four Co
atoms with Cu the energy rises above the convex hull. In contrast, we
can replace any number of Co atoms with Ni and the structure remains
on the convex hull. Thus our calculation shows the solubility of Ni in
\alco{9}{2} extending over the entire line $x+y=0.1818$.
In reality (see Fig.~\ref{fig:ternaries}b), stability of
Al$_9$(Co,Ni)$_2$ terminates after about half of the Co are replaced
with Ni. It is likely that in a full GPT treatment including the
atomic volume and composition dependence of the potentials this
feature would be accurately captured, because it is known that the
\alco{9}{2} structure lies slightly above the convex hull of the
\alni{1-y}{y} binary alloy when the full GPT potentials are employed.
Still, the trend of greater Ni solubility than Cu in
\alco{9}{2} is faithfully reproduced.

Along $x+y=1/4$ we study the ability of Co to replace Ni in the
\dosb{11} structure of \alni{3}{}. We find that Co can fully replace
Ni within this structure.  Experimentally, up to half the Ni can be so
replaced before a competing phase O-Al$_{13}$(Co,Ni)$_4$ appears.
According to our calculations, and consistently with experiment, the
same \dosb{11} structure is not stable for \alcu{3}{} nor is it stable
with limited amounts of Co replacing Cu. Again, the ability of Co and
Ni to replace each other within their binary structures is faithfully
reproduced, as is the relative inability of Co and Cu to replace each
other in the same structures.

\section{Quasicrystal}
\label{sec:QC}

The hope to model quasicrystal structures was an important
motivation for the development of these ternary GPT potentials. We
have calculated the stability and cohesive energy of the decagonal
quasicrystal phase in Al-Co-Cu. As a structural model for this phase
we employ the model of Cockayne and Widom~\cite{CW98}. This model was
deduced from Monte Carlo simulations of the alloy using a non-rigorous
total energy calculation based upon ``mock-ternary'' interactions.
Fortunately, in this structure there are no near-neighbor Co atoms.
Thus, the Co-Co overbinding at short distances will not be problematic
and we may directly apply the GPT potentials.

According to our present calculations, this model quasicrystal enjoys
both modest mechanical and thermodynamic stability. The median atomic
displacement of 0.14~\AA~ and mean atomic displacement of
0.18~\AA~ is encouraging.  However, a few Al atoms
displace by over 1~\AA.  Such behavior has been found previously
in~\cite{CW97,CW98,MarekandSuck}. We attribute these large
displacements to nearly flat potential energy surfaces around certain
atomic positions that allow small errors in the forces to cause large
atomic displacements. The thermodynamic stability of the decagonal
phase is also limited. The quasicrystal phase lies only 37 meV/atom
below the tie-line with $\tau'$, M-(Al,Cu)$_{13}$Co$_4$ and
\alcocu{2}{}{}. A more exhaustive consideration of hypothetical
structures, including other quasicrystal approximants and other
crystalline structures, could possibly find structures that compete in
energy with the quasicrystal.

Favorable bond lengths contribute to the low energy of our
quasicrystal model. Pair correlation functions $g_{\alpha\beta}(r)$
between atomic species $\alpha$ and $\beta$ generally exhibit maxima
near to separations $r$ at which the potential $v_2^{\alpha\beta}(r)$
exhibits minima, and vice versa. This relationship is especially
notable for AlCo and CoCo pairs, as illustrated Fig.~\ref{fig:gofr}.
The correlation functions in this figure incorporate gaussian
broadening to mimic the effects of thermal fluctuations at T=1000K.
All of the AlCo near neighbors separations lie within the first deep
minimum of the AlCo pair potential around 2.4~\AA. The second pair
potential minimum at 4.4~\AA, and the third minimum at 6.4~\AA
likewise encompass strong second and third peaks of the pair
correlation functions. The first minimum of the CoCo pair potential
occurs at an unphysically short separation. Fortunately, the CW model
includes no CoCo pairs at this separation. However, the strong second
and third CoCo pair potential minima, at 4.5~\AA~ and 6.5~\AA
respectively, match the first two peaks of the CoCo pair correlation
function.

A side effect of the favorable bonding environments of Co atoms is the
existence of a highly stable Co network within the decagonal phase
model. When the mean atomic displacement under relaxation is broken
down according to atomic species we find mean displacements of
0.22~\AA~ for Al atoms, 0.15~\AA~ for Cu atoms and 0.08~\AA~ for Co
atoms.  Figure~\ref{fig:decag} illustrates a portion of a decagonal
AlCoCu structure and highlights the strong CoCo bonds. In addition to
the bonds drawn, there are favorable near neighbor separations
involving Al and Co atoms as well as further neighbor separations. It
is evident that the geometry of the decagonal phase structure exploits
the oscillations of the interatomic pair potentials to achieve a low
energy. The bonds illustrated in Fig.~\ref{fig:decag} form edges of a
pentagon-rectangle-triangle tiling, a well known motif in decagonal
quasicrystals and approximants.

We are currently engaged in full ab-initio calculations of total
energies in small unit cell AlCoCu~\cite{WidLSMS} and
Al-Co-Ni~\cite{IbrahimMGPT} quasicrystal approximants.  The
information thus obtained will allow us to modify the TM interactions
at short distances, formally including many-body interactions within
the pair potential~\cite{Moriarty90}. The resulting interactions
become structure dependent because they are tuned for atomic
environments typical of the quasicrystal. They should allow modeling
of the quasicrystal phase at the pair potential level. Indeed, the
structure displayed in Fig.~\ref{fig:decag} results from a Monte Carlo
simulation~\cite{ThankMarek} that takes sites characteristic of the
Cockayne and Widom decagonal phase model~\cite{CW98} and anneals the
chemical occupation of those sites using modified GPT potentials. The
structures that result are in close agreement with the predictions of
Cockayne and Widom~\cite{CW98} based upon mock ternary potentials.

\section{Conclusions}

We have developed first-principles pair potentials for Al-Co-Cu
and Al-Co-Ni alloys. These potentials reproduce many
features of the known phase diagrams, placing known stable and
metastable structures on or near the convex hull of energy versus
composition plots. Comparisons of the sequence of binary alloys Al-Cu,
Al-Ni and AlCo, and comparisons of the ternary alloys Al-Co-Cu and
Al-Co-Ni, accurately reflect the variations in phase diagrams
among these compounds.

The range of applicability of the AlCoCu potentials
extends up to total transition metal concentration of 50\%
provided the Co concentration is below 25\%. The range of validity
includes the composition of a decagonal quasicrystal phase. Even
within this range, however, small errors are observed in the
spurious appearance of a presumed metastable $\tau$ phase on the
convex hull, and the calculated metastability of a known stable
phase \alcocu{7}{}{2}.

Much more serious are the difficulties at larger Co concentration.
Here, both the assumption of constant electron density and the
neglect of many-body interactions prohibit application of these
potentials.


\acknowledgments MW acknowledges support by the National Science
Foundation under grant DMR-9732567. IA is supported by a
fellowship of the Saudi Arabian government. The work of the JAM
was performed under the auspices of the U.S.  Department of Energy
by the Lawrence Livermore National Laboratory under contract
number W-7405-ENG-48.


\begin{references}

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\bibitem{QCReviews}
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M. Windisch, M. Krajci and J. Hafner, J. Phys: Cond. Matt. {\bf 6},
6977 (1994);
%Electronic structure of AlCoCu
G. Trambly de Laissardiere and T. Fujiwara,
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G. Trambly de Laissardiere, D. Nguyen Manh, L. Magaud, J.~P. Julien,
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%Electronic structure of AlCoCu
R.~F. Sabiryanov, S.~K. Bose and S.~E. Burkov, J. Phys: Cond. mat. {\bf 7}
5437 (1995);
%Electronic structure of AlCoCu
M. Krajci, J. Hafner and M. Mihalkovic, Phys. Rev. B {\bf 56}
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%Early Al GPT results
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J.~A. Moriarty and A.~K. McMahan, Phys. Rev. Lett. {\bf 48}, 809 (1982);
A.~K. McMahan and J.~A. Moriarty, Phys. Rev. B {\bf 27}, 3235 (1983).

\bibitem{Moriarty88}
J.~A. Moriarty, Phys. Rev. B {\bf 38}, 3199 (1988).

%AlMgZn pair potentials
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(1992)

%AlCo potentials
\bibitem{Phillips94}
R. Phillips, J. Zou, A.~E. Carlsson, and M. Widom, Phys. Rev. B {\bf 49},
9322  (1994).

%AlCuLi pair potentials
\bibitem{Windisch94pot} M. Windisch, J. Hafner, M. Krajci and M. Mihalkovic,
Phys. Rev. B {\bf 49}, 8701 (1994)

%AlMn potentials
\bibitem{Mihalk}
M. Mihalkovic, W.-J. Zhu, C.~L. Henley, and R. Phillips, Phys. Rev. B
{\bf 53}, 9021 (1996).

\bibitem{paper_I}
J.~A. Moriarty and M. Widom, Phys. Rev. B {\bf 56}, 7905 (1997).

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M. Widom and J.~A. Moriarty, Phys. Rev. B {\bf 58}, 8967 (1998).

\bibitem{MeltingMo} J.A. Moriarty, Phys. Rev. B {\bf 49}, 12431 (1994).

%Max Born proceedings - V(r) compared with g(r)
\bibitem{WC96}
M. Widom and E. Cockayne, Physica A {\bf 232}, 713 (1996).

%Ortho structure
\bibitem{Grin}
J. Grin, U. Burkhardt, M. Ellner, and K. Peters, J. Alloys and Compounds
{\bf 206}, 243 (1994).

%C2/m structure for Al13Co4
\bibitem{Freiburg}
C. Freiburg, B. Grushko, R. Wittenberg, and W. Reichert, Mat. Sci. Forum
{\bf 228-231}, 583 (1996).

%Complete AlCu evaluation
\bibitem{JLMurray} J.~L. Murray, Int. Met. Rev. {\bf 30}, 211 (1985)

%AlCu (LT)
\bibitem{Boragy} M. El-Boragy, R. Szepan and K. Schubert,
J. Less-Common Metals, {\bf 29} 133 (1972)

%AlCu (HT)
\bibitem{Preston}G.~D. Preston, Phil. Mag. {\bf 12} 980 (1931)

%Cm structure for Al13Co4
\bibitem{Hudd}
R.~C. Hudd and W.~H. Taylor, Acta Cryst. {\bf 15}, 441 (1962).

%tau phase references
\bibitem{tau} S.~S. Lu and T. Chang, Acta Sci. Sin. {\bf 13}, 150 (1957);
G. van Tendeloo, C. van Heurck and S. Amelinckx, Sol. State Comm. {\bf 71},
705 (1989)

%tau phases in Al-Cu
\bibitem{Sastry} G.~V.~S. Sastry, K.~K. Bariar, K. Chattopadhyay and P.
Ramachandrarao, Z. Metallk. {\bf 71}, 756 (1980)

%Vacancy ordered phases
\bibitem{Chattopadhyay}
K. Chattopadhyay, S. Lele, N. Thangaraj and S. Ranganathan,
Acta Metall. {\bf 35} 727 (1987)

%Metastable tau sequence in AlCoCu.
\bibitem{He88}
L.~X. He, X.~Z. Li, Z. Zhang and K.~H. Kuo, J. Mat. Sci. Lett. {\bf 7},
1284 (1988)

%Isothermal sections of AlCoNi phase equilibria
\bibitem{Godecke}
T. G\"{o}decke, M. Scheffer, R. L\"{u}ck, S. Ritsch and C. Beeli,
Z. Metallkd. {\bf 89}, 687 (1998)

%PD of Al-Co-Cu
\bibitem{Gru93}
B. Grushko, Phase Trans. {\bf 44}, 99 (1993)

% tP40 structure
\bibitem{Bown}
M.G. Bown and P.J. Brown, Acta Cryst. {\bf 9}, 911 (1956)

% early D- AlCoCu reports
\bibitem{D-AlCoCu}
L.X. He, Y.K. Wu and K.H. Kuo, J. Mater. Sci. Lett. {\bf 7}, 1284 (1988);
A.-P. Tsai, A. Inoue and T. Masumoto, Mater. Trans. JIM, {\bf 30}, 300 (1989)


\bibitem{Raynor} G.~V. Raynor, Prog. Met. Phys. {\bf 1}, 1 (1949)

%e/a ratio of AlCoCu, AlCoNi and AlNiFe
\bibitem{eoveraratio}
B. Grushko and K. Urban, Phil. Mag. B {\bf 70}, 1063 (1994)

%Vacancies in Al(Co,Cu)
\bibitem{Ridley}
N. Ridley, J. Inst. Met. {\bf 94}, 255 (1966).

%Comparisons of Al-Co-Cu with Al-Co-Ni PD
\bibitem{Gru94}
B. Grushko in {\em Aperiodic '94}, eds G. Chapuis and W. Paciorek 
(World Scientific, 1995)

%(Al,Cu)13Co4 structure
\bibitem{FG}
C. Freiburg and B. Grushko, Z. Krist. {\bf 209}, 49 (1994)

\bibitem{CW97}
E. Cockayne and M. Widom, Phil. Mag. A {\bf 77}, 593 (1998)

\bibitem{CW98} E. Cockayne and M. Widom, Phys. Rev. Lett.

%Large displacements in D-phase relaxation
\bibitem{MarekandSuck} 
M. Mihalkovic, F. Dugain and J.~B. Suck, J. Non-Cryst. Sol.
{\bf 205-207}, 701 (1996)

%LSMS studies of AlCoCu
\bibitem{WidLSMS} M. Widom, I. Al-Lehyani, Y. Wang and E. Cockayne,
Mat. Sci. Eng. A, to appear (2000)

%VASP studies of AlCoNi
\bibitem{IbrahimMGPT} I. Al-Lehyani, N. Moghadam, J.~A. Moriarty,
Y. Wang and M. Widom, in preparation (1999)

\bibitem{ThankMarek} We are indebted to Marek Mihalkovic for developing
the computer program used in this simulation.

%folding v3 and v4 into v2.
\bibitem{Moriarty90}
J.~A. Moriarty, Phys. Rev. B {\bf 42}, 1609 (1990) and {\bf 49}, 12 431
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\bibitem{Tsai87}
A.~P. Tsai, A. Inoue, and T. Masumoto, Jpn. J. Appl. Phys. {\bf 26},
L1505 (1987).

\bibitem{Pearson}
P. Villars and L.~D. Calvert, {\it Pearson's Handbook of
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%Al75Co22Ni3
\bibitem{ZGS}
B. Zhang, V. Gramlich, and W. Steurer, Zeit. Krist. {\bf 210},
498 (1995).

\end{references}


\begin{table}
\caption{Structural data for real and hypothetical Al-Cu
phases up to $y=0.5$}
\begin{tabular}{|l|l|l|l|l|}
\hline
Name& \parbox{1.5in}{Pearson symbol (Strukturebericht)} & Space
group & Reference & $y$ \\
\hline

FCC         & cF4 (A1) &Fm$\bar{3}$m & \cite{Pearson} & 0.0000 \\

HCP         & hP2  (A3)     & P6$_3$/mmc & \cite{Pearson} & 0.0000 \\

BCC         & cI2  (A2)     & Im3m       & \cite{Pearson} & 0.0000 \\

Al$_{12}$W  & cI26          & Im3        & \cite{Pearson} & 0.0769 \\

\almn{6}{}  & oC28 (D2$_h$) & Cmcm       & \cite{Pearson} & 0.1429 \\

\alco{9}{2} & mP22          & P2$_1$/a   & \cite{Pearson} & 0.1818 \\

O-\alco{13}{4}& oP102       & Pmn2$_1$   & \cite{Grin}    & 0.2353 \\

\alconi{75}{22}{3} & mC34-1.8  & C2/m    & \cite{ZGS}     & 0.2500 \\

W$_3$O      & cP8 (A15)     & Pm3n       & \cite{Pearson} & 0.2500 \\

AuCu$_3$    & cP4 (\lsb)  & Pm3m       & \cite{Pearson} & 0.2500 \\

Fe$_3$Al    & cF16 (\dosb{3})  & Fm3m    & \cite{Pearson} & 0.2500 \\

Al$_3$Ti    & tI8 (\dosb{22})  & I4/mmm  & \cite{Pearson} & 0.2500 \\

\alni{3}{}  & oP16 (\dosb{11}) & Pnma    & \cite{Pearson} & 0.2500 \\

M-\alco{13}{4}& mC102       & C2/m       & \cite{Pearson,Freiburg} & 0.2637 \\

\alco{5}{2} & hP28          & P6$_3$/mmc & \cite{Pearson} & 0.2857 \\

\alfe{5}{2} & oC16          & Cmcm       & \cite{Pearson} & 0.2857 \\

\alcocu{7}{}{2} & tP40      & P4/mnc     & \cite{Pearson} & 0.3000 \\

Al$_2$Cu    & tI12 (C16)    & I4/mcm     & \cite{Pearson} & 0.3333 \\

Al$_7$Cu$_4$Li & cF12       & Fm$\bar{3}$m & \cite{Pearson} & 0.3333 \\

Al$_4$CoLa  & oP12          & Pmma       & \cite{Pearson} & 0.3333 \\

$\tau_3$ (\alcu{3}{2}) & hP5 (D5$_{13}$) & P$\bar{3}$m1 
      & \cite{Pearson} & 0.4000 \\

$\tau_n$    &               &         &\cite{Chattopadhyay} & 0.375-0.400 \\

\alconi{4}{}{2} & cI112     & Ia$\bar{3}$d & \cite{Pearson} & 0.4286 \\

CsCl        & cP2 (B2)      & Pm3m      & \cite{Pearson} & 0.5000 \\

NaCl        & cF8 (BA4)     & Fm3m      & \cite{Pearson} & 0.5000 \\

AuCu        & tP4 (L1$_0$)  & P4/mmm    & \cite{Pearson} & 0.5000 \\

AlCu (HT)   & oC16          & Cmmm      & \cite{Preston} & 0.5000 \\

AlCu (LT)   & mC20          & C2/m      & \cite{Pearson,Boragy} & 0.5000 \\

\end{tabular}
\label{tab:alcu}
\end{table}

\newpage
\begin{table}
\caption{Structural data for real and hypothetical Al-Co-Cu
phases up to $x+y=0.5$}
\begin{tabular}{|l|l|l|l|l|}
Name& Pearson symbol & Space group & 
$(x,y)$ \\
\hline

Al$_{12}$(Cr,Mn) & cI26 & Im$\bar{3}$ & (0.0385,0.0385) \\

\parbox{1in}{Al$_{11}$CoMo$_4$\\} & \parbox{1in}{tI8\\}   &
\parbox{1in}{I4/mmm\\}      & \parbox{1in}{(0.0625,0.1875)\\
                                             (0.1875,0.0625)} \\

\parbox{1in}{Al$_{11}$CoMo$_4$\\} & \parbox{1in}{tI8\\}   &
\parbox{1in}{I4/mmm\\} & \parbox{1in}{(0.0625,0.1875)\\
                                      (0.1875,0.0625)} \\

Al$_6$MoNb & tI8 & I4/mmm & (0.1250,0.1250) \\

Al$_{23}$CuFe$_4$ & cO28 & Cmc2$_1$ & (0.1429,0.1071) \\

Al$_4$CoLa  & oP12  & Pmma    & (0.1667,0.1667) \\

Al$_4$NiY & oC24 & Cmcm & (0.1667,0.1667) \\

\parbox{1in}{Al$_8$La$_3$Sc\\} & \parbox{1in}{cF24\\} &
\parbox{1in}{Fd$\bar{3}$m\\} & \parbox{1in}{(0.2500,0.0833)\\
                                                     (0.0833,0.2500)} \\

Al$_4$LaY & cF24 & Fd$\bar{3}$m & (0.1667,0.1667)  \\

\parbox{1in}{Al$_8$NiTi$_3$\\} & \parbox{1in}{cP4\\} &
\parbox{1in}{Pm$\bar{3}$m\\} & \parbox{1in}{(0.2500,0.0833)\\
                                              (0.0833,0.2500)}\\

\parbox{1in}{Al$_5$CuHf$_2$\\} & \parbox{1in}{cP4\\} &
\parbox{1in}{Pm$\bar{3}$m\\} & \parbox{1in}{(0.1250,0.2500)\\
                                              (0.2500,0.1250)} \\

Al$_3$CuHo & oI10 & Immm & (0.2000,0.2000) \\

Al$_2$FeNi & cP2 & Pm$\bar{3}$m & (0.2500,0.2500) \\

Al$_2$HfZn & cP4 & Pm$\bar{3}$m & (0.2500,0.2500) \\

Al$_2$NiY & oC16 & Cmcm & (0.2500,0.2500) \\

Al$_2$CoY   & oC16  & Cmcm & (0.2500,0.2500) \\

\end{tabular}
\label{tab:alcocu}
\end{table}

\newpage

\begin{figure}
%\epsfig{figure=AlCoCu.eps,width=6in}
%\epsfig{figure=AlCoCu.eps,width=6in}
\vspace{10pt}
\caption{Ternary phase diagrams of (a) Al-Co-Cu (adapted from 
Refs.~\cite{Gru93} and~\cite{Gru94}) and (b) Al-Co-Ni (adapted from
Ref.~\cite{Godecke}).}
\label{fig:ternaries}
\end{figure}

%\newpage
\begin{figure}
%\epsfig{figure=Alpot.eps,width=6in}
%\epsfig{figure=TMpot.eps,width=6in}
\caption{Interatomic pair potentials calculated in the limit
$x,y \rightarrow 0$. (a) Aluminum interactions. (b) Transition metal
interactions}
\label{fig:potentials}
\end{figure}

%\newpage
\begin{figure}
%\epsfig{figure=Al(1-x)Cux.ps,width=6in}
\caption{Scatter plot of binary \alcu{1-y}{y} structural energies $E(y)$.
Plotting symbols indicate displacements under relaxation: 
unrelaxed symmetric structure ($\boxplus$);
$\Delta R \le 0.1$~\AA ($\bullet$);
0.1~\AA $< \Delta R \le 0.2$~\AA ($\blacktriangle$);
0.2~\AA $< \Delta R \le 0.3$~\AA ($\blacksquare$);
unrelaxed unstable structure ($\square$). The solid line indicates the
convex hull.}
\label{fig:Eofy}
\end{figure}

%\newpage
\begin{figure}
%\epsfig{figure=AlCoCu.eps,width=6in}
\caption{Al-Co-Cu ternary phase diagram. Labeled circles denote
experimentally observed phases. Triangles denote other compositions
considered. Solid lines border facets of convex hull in calculated
$E(x,y)$.}
\label{fig:alcocu}
\end{figure}

%\newpage
\begin{figure}
%\epsfig{figure=Al(1-2x)CoxCux.ps,width=6in}
\caption{Scatter plot of ternary structural energies $E(x,y)$ along the line
$x=y$. Plotting symbols as in Fig.~\ref{fig:Eofy}}
\label{fig:Eofx=y}
\end{figure}

%\newpage
\begin{figure}
%\epsfig{figure=Al.75Co.25-yCuy.ps,width=6in}
\caption{Scatter plot of ternary structural energies $E(x,y)$ along the line
$x+y=1/4$. Plotting symbols as in Fig.~\ref{fig:Eofy}}
\label{fig:Eofx+y=1/4}
\end{figure}

%\newpage
\begin{figure}
%\epsfig{figure=Al.67Co.33-yCuy.ps,width=6in}
\caption{Scatter plot of ternary structural energies $E(x,y)$ along the line
$x+y=1/3$. Plotting symbols as in Fig.~\ref{fig:Eofy}}
\label{fig:Eofx+y=1/3}
\end{figure}

\begin{figure}
\caption{Cobalt correlation functions in the decagonal phase of Al-Co-Cu
exhibit maxima near minima of the pair potentials.  (a) Al-Al potential
and correlation function. (b) Co-Co potential and correlation function.}
\label{fig:gofr}
\end{figure}

\begin{figure}
\caption{Structure of the AlCoCu decagonal phase. Open circles indicate
Al atoms, gray disks Cu atoms and black disks Co atoms. Large and
small atoms occur in parallel layers 2~\AA~ apart. Gray and Black
bonds joining neighboring Co atoms are 4.5 and 4.7~\AA, respectively.}
\label{fig:decag}
\end{figure}


\end{document}
