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\newcommand{\altm}[2]{\mbox{Al$_{#1}$TM$_{#2}$}}
\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{\dzerosb}[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 structural and mechanical properties of intermetallic compounds
and alloys 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 transition-metal (TM) aluminides 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 phase diagrams of the binary alloy Al-Cu and 
the ternary systems Al-Co-Cu and Al-Co-Ni, using GPT pair potentials 
calculated in the limit of vanishing transition-metal concentration. 
Despite this highly simplifying approximation, we find rough
agreement with the known low-temperature phase diagrams, up to 50\% 
total TM concentration provided the Co fraction is below 25\%.  Full 
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 cohesion and atomic interactions 
govern crystal structure and important mechanical properties such
as elastic constants of solids, understanding the interactions
among atoms within intermetallic 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. {\it Ab-initio}
electronic-structure methods applied to low-symmetry intermetallic
structures~\cite{AbInitio,Krajci} can become computationally very
demanding, due to large unit cells with many inequivalent atomic sites
and the likelihood of structural and chemical disorder.
Quantum-mechanically-based interatomic potentials
\cite{Moriarty8283,Moriarty88,Krajci92,Phillips94,Windisch94pot,Mihalk},
on the other hand, are well suited to complex or disordered structures. 
At the same time, the multiplicity of chemical species requires the 
calculation of numerous interaction potentials, which may be composition
dependent. The presence of transition-metal (TM) components may 
require that angular-dependent many-body interactions be considered if 
the TM concentrations are sufficiently high.  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 localized orbitals for TM $d$ electrons to calculate 
such potentials from first principles.

Previous papers (henceforth referred to as paper I~\cite{paper_I} and 
paper II~\cite{paper_II}) extended the formalism of the first-principles 
generalized pseudopotential theory (GPT) for elemental interatomic 
potentials~\cite{Moriarty88} to aluminum-rich \altm{1-x}{x} binary 
intermetallics, and studied their applicability to the systems Al-Co 
and Al-Ni in detail.  This paper further extends the GPT approach 
to ternary intermetallic alloys, specifically Al-Co-Cu and Al-Co-Ni. 
Previously calculated ternary alloy potentials~\cite{Krajci92,Windisch94pot} 
have treated only simple metals, without $d$-electron interactions, 
and noble metals, with assumed completely filled $d$ shells. The present 
paper is thus the first treatment of interatomic potentials in ternary 
alloys containing transition metals with partially filled $d$ bands.

Our goal in the present paper is to test the ability of the GPT
interatomic pair potentials, calculated in the zero TM-concentration 
limit, to reproduce the experimentally known aluminide 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 interatomic 
interactions.

Our work focuses on the ternary system 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 low-temperature phase diagram, 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 possible structures must lie above this convex hull.  We cannot,
of course, 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 Sec.~\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 at the present level of approximation
\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 alloy phase diagram of Al-Cu, followed by a treatment
of ternary phase diagrams in Sec.~\ref{sec:ternary}. In 
Sec.~\ref{sec:QC} we discuss the utility of these potentials
applied to decagonal quasicrystal structures, and we conclude in
Sec.~\ref{sec:conclusions}.


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

Paper I described the theoretical basis for calculating interatomic 
potentials in binary alloys within the generalized pseudopotential 
theory.  Here, we review and extend some key ideas.  For a general 
multi-component alloy, the GPT interatomic potentials are explicit 
terms in a real-space expansion of total energy, which takes the 
form of a collective volume term, central-force pair interactions, 
and angular-force 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.  The quantity $\Omega$ is the average
atomic volume in the alloy and {\bf c} is a composition vector 
whose elements $x,y, \cdot\cdot\cdot$ depend upon the concentrations 
of the 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 ion sites 
occupied by the corresponding species.

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 ($v^{\alpha 
\beta}_2$) are the leading structure-dependent terms in the total 
energy.  The many-body interactions ($v^{\alpha \beta \gamma}_3$ 
and $v^{\alpha \beta \gamma \delta}_4$) are presumed to be 
strongest among clusters of transition-metal atoms, due to the 
directional bonding of their $d$ electrons, and weaker among 
clusters containing simple-metal (e.g., aluminum) atoms.  
Consequently, the many-body interactions should be negligible at 
low TM concentrations, and grow progressively more important at 
higher TM concentrations.  In pure elemental transition metals, 
the three- and four-body interactions are important, although 
higher-order interactions may often be neglected 
\cite{Moriarty88,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 strong correlations 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.  Papers I and II discussed the 
detailed first-principles evaluation of full volume- and 
composition-dependent potentials for binary \altm{1-x}{x} alloys.  
The extension of those procedures to ternary systems is reasonably 
straigthforward, but it is clearly quite burdensome in practice when 
so many volumes and compositions are involved, as is the case here.  
For this reason, the discussion in the remainder of the paper uses 
only the GPT pair potentials evaluated in the limit of vanishing 
transition-metal concentration ($x=y=0$) and applied under the 
assumption of constant valence electron density, with the volume 
term treated as a constant.  These simplifications are 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 have confirmed in paper II that the limiting $x=0$ 
potentials so applied achieve considerable success, although we did 
find that a few details in the alloy phase diagrams require the 
full volume- and composition-dependent GPT, and for $x > 0.264$
many-body potentials as well.  Nonetheless, the simplicity and
elegance of the limiting aluminum-rich GPT treatment makes it the 
logical starting point for a consideration of ternary aluminide 
phase diagrams.

Calculated pair potentials for the Al-Co-Cu and Al-Co-Ni systems,
computed in the zero TM-concentration 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 pair interactions of the
same transition-metal atoms.  In the present work, 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. Quantitatively, the magnitude of $v_2^{\rm NiNi}-(v_2^{\rm
CoCo}+v_2^{\rm CuCu})/2$ does not exceed 0.03 eV for $r \ge 2.5$~\AA.
One final point to note in Fig.~\ref{fig:potentials}b is the apparent
strong binding of Co-Co pairs at unphysically short distances.  This
feature is a known difficulty of the unbalanced 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 term 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 strong unbalanced transition-metal
pair 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 GPT pair 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 
volume- and composition-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 three- and four-body TM 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 \dzerosb{11} 
structure of \alni{3}{}. Both of these difficulties were alleviated 
by use of the volume- and composition-dependent potentials.


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

Since Cu is a noble metal, with a completely filled $d$ shell in the
atom and nearly filled $d$ bands in the elemental metal, its
many-body interactions are relatively weak compared with those
among other transition-metal atoms.  That suggests that the GPT
should apply to the compound \alcu{1-y}{y} for all $y$ from 0 to 1,
keeping only the volume term and the pair 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 
in the present work and because we employ pair potentials calculated 
in the limit of vanishing Cu concentration.  In this limit, Cu has
a calculated $sp$ valence of 1.805, compared to a value of 1.651 in
the elemental metal and a value of 1.0 in the free atom.

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 \dzerosb{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 Al-Co-Ni and Al-Co-Cu
are known in some detail~\cite{Godecke,Gru93}. The labeled regions in
Fig.~\ref{fig:ternaries} represent the known stable phases.
Al-Co-Cu (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 Al-Cu 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'' Al-Co-Cu
potentials~\cite{CW98} that started with binary Al-Co
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 Al-Co-Ni 
(Fig.~\ref{fig:ternaries}b) contrasts strongly~\cite{Gru94} with
Al-Co-Cu.  Instead of well isolated composition fields, the phase
diagram of Al-Co-Ni 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 Al-Co-Cu. 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 in Fig.~\ref{fig:alcocu} 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 (denoted
``D'' in Fig.~\ref{fig:alcocu}).  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 (denoted ``M'' in fig.~\ref{fig:alcocu}) 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} which is
close to the experimentally reported composition. A simulated
annealing procedure~\cite{paper_II} 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 Al-Cu 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 in Sec.~\ref{sec:totE} 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 in
Figs.~\ref{fig:Eofx=y}-~\ref{fig:Eofx+y=1/3}.  As noted above,
\alcocu{7}{}{2} lies slightly above the convex hull, and thus
tie-lines connecting to it (see $\times$ symbols) lie slightly above
the hull. 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 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. We did not consider these binaries as part of our ternary
study because at this composition a more careful treateent of vacancy
formation is needed. 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 interatomic potentials to reproduce trends among
distinct compounds is a further test of their applicability. We
examined two special lines in the Al-Co-Ni phase diagram and
contrasted them with the same lines in the Al-Co-Cu diagram.  Our goal
is to understand the differing abilities of transition metals to
substitute for each 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. This is
consistent with the experimental phase diagram
(Fig.~\ref{fig:ternaries}a) in which Cu is insoluble in \alco{9}{2}.
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.
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
\dzerosb{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 \dzerosb{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 motivated the development of
these ternary GPT potentials. We 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. Note that this model is one of many competing decagonal
phase models (see Ref.~\cite{Krajci} and references therein). We do
not attempt to distinguish among competing models in the present paper
because a more thorough treatment is transition-metal binding would be
required than is possible within our present approximations.

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 Refs.~\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 $\tau_3$, 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 Al-Co and Co-Co pairs, as illustrated Fig.~\ref{fig:gofr}.
The correlation functions in this figure incorporate gaussian
broadening to mimic the effects of thermal fluctuations at $T=1000$~K.
Without broadening the correlation functions would be a dense collection
of closely-spaced $\delta$-functions.

All of the Al-Co near neighbors separations lie within the first deep
minimum of the Al-Co 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 Co-Co pair potential occurs at 
an unphysically short separation.  Fortunately, the CW model includes 
no Co-Co pairs with separations below 4~\AA.  However, the strong second and 
third Co-Co pair-potential minima, at 4.5~\AA~and 6.5~\AA~respectively, 
match the first two peaks of the Co-Co 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
Al-Co-Cu structure and highlights the strong Co-Co bonds. In addition
to the bonds drawn, there are favorable Al-Co and Co-Cu near-neighbor
separations, as well as favorable further neighbor separations. 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. It is evident that the geometry of the
decagonal phase structure exploits the oscillations of the interatomic
pair potentials to achieve a low energy.

We are currently engaged in {\it ab-initio} electronic-structure 
calculations of total energies in small-unit-cell Al-Co-Cu~\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 TM pair potentials~\cite{Moriarty90}. The resulting pair 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 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 based upon mock ternary potentials.


\section{Conclusions}
\label{sec:conclusions}

We have extended the generalized pseudopotential theory of interatomic
potentials in binary transition-metal aluminides to ternary systems,
and have developed first-principles pair potentials in the
aluminum-rich limit for Al-Co-Cu and Al-Co-Ni alloys.  The pair
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. The known stable and
metastable structures also exhibit mechanical stability under static
relaxation. Comparisons of the sequence of binary alloys Al-Cu, Al-Ni
and Al-Co, 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 present Al-Co-Cu pair potentials
extends up to a 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}.

More serious are the difficulties at larger Co concentrations.
Here, both the assumption of constant electron density and the
neglect of many-body interactions limit the useful application of 
the zero TM-concentration pair potentials.  As previously found for 
\alco{1-x}{x}, full volume- and concentration-dependent GPT pair 
potentials and/or corresponding three- and four-body Co potentials 
are needed to accurately address this regime.


\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.


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

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

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J. Grin, U. Burkhardt, M. Ellner, and K. Peters, J. Alloys and Compounds
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K. Chattopadhyay, S. Lele, N. Thangaraj and S. Ranganathan,
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B. Grushko and K. Urban, Phil. Mag. B {\bf 70}, 1063 (1994).

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B. Grushko in {\em Aperiodic '94}, eds G. Chapuis and W. Paciorek 
(World Scientific, 1995).

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C. Freiburg and B. Grushko, Z. Krist. {\bf 209}, 49 (1994).

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

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{\bf 205-207}, 701 (1996).

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\bibitem{ThankMarek} We are indebted to Marek Mihalkovic for developing
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\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 (\dzerosb{3})  & Fm3m    & \cite{Pearson} & 0.2500 \\

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

\alni{3}{}  & oP16 (\dzerosb{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 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.~\protect\cite{Gru93} and~\protect\cite{Gru94}) and (b) Al-Co-Ni
(adapted from Ref.~\protect\cite{Godecke}). Structural information is
listed in Table~\ref{tab:alcu} for all phases except for ``D''
(decagonal, Ref.~\protect\cite{D-AlCoCu}) and ``X'' (unknown
triclinic, Ref.~\protect\cite{X})}
\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 $\Delta E(y)$.
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. Dashed lines border facets of convex hull in calculated
$\Delta E(x,y)$.}
\label{fig:alcocu}
\end{figure}

%\newpage
\begin{figure}
%\epsfig{figure=Al(1-2x)CoxCux.ps,width=6in}
\caption{Scatter plot of ternary structural energies $\Delta E(x,y)$ 
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 $\Delta E(x,y)$ 
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 $\Delta E(x,y)$ 
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 Al-Co-Cu 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}
