8a9
> \newcommand{\altm}[2]{\mbox{Al$_{#1}$TM$_{#2}$}}
41,43c42,44
< 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);
---
> 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);
45,55c46,57
< 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
---
> 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 tolerable 
> 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
58,61c60,63
< 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
---
> 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
77c79
< scientific interest. Because atomic interactions and cohesion
---
> scientific interest. Because cohesion and atomic interactions 
84,97c86,101
< 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.
---
> is a considerable theoretical challenge. {\it Ab-initio} electronic-structure
> methods applied to low-symmetry intermetallic structures~\cite{AbInitio} 
> 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, and 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.
99,110c103,114
< 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.
---
> 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.
113,120c117,125
< 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.
---
> 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.
122,126c127,131
< 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
---
> 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
140,146c145,152
< 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.
---
> 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.
150c156
< interest. Along the the binary \alcu{1-y}{y} axis we find nearly
---
> interest.  Along the the binary \alcu{1-y}{y} axis we find nearly
152,160c158,166
< 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
---
> 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
162,167c168,173
< \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.
---
> \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.
172,175c178,182
< 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.
---
> 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}.
176a184
> 
180,184c188,194
< 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:
---
> 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:
202,204c212,215
< 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,
---
> 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,
206,207c217,218
< $i, j, k, l, \cdot\cdot\cdot$ run over the individual ions $1,
< \cdot\cdot\cdot, N$.
---
> $i, j, k, l, \cdot\cdot\cdot$ run over the individual ion sites 
> occupied by the corresponding species.
209,227c220,240
< 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}.
---
> 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}.
230,237c243,263
< 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.
---
> 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 
> phases diagrams.
240,248c266,273
< 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.
---
> 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.
250,265c275,290
< 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.
---
> 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.  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.
270,271c295,296
< back to the approximations employed. First of all, taking the
< limit of vanishing transition metal concentration, and making the
---
> back to the approximations employed.  First of all, taking the
> limit of vanishing transition-metal concentration, and making the
273c298
< of volume energies and pair potentials with composition and atomic
---
> of volume term and pair potentials with composition and atomic
278,281c303,306
< 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.
---
> 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.
285,298c310,323
< 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.
---
> 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 \dosb{11} 
> structure of \alni{3}{}. Both of these difficulties were alleviated 
> by use of the volume- and composition-dependent potentials.
299a325
> 
303,307c329,334
< 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
---
> 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
309,311c336,340
< $y=1/2$, because this is the concentration range that interests us at
< present and because we employ potentials calculated in the limit
< of vanishing Cu concentration.
---
> $y=1/2$, 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.
320c349
< low-temperature (LT) structure~\cite{Boragy} mC20 and a high
---
> low-temperature (LT) structure~\cite{Boragy} mC20 and a high-
364c393
< in units of $a_0$. We introduce a pair of Al vacancies into one layer
---
> in units of $a_0$.  We introduce a pair of Al vacancies into one layer
369c398
< For each of the structures listed in table~\ref{tab:alcu} we
---
> For each of the structures listed in Table~\ref{tab:alcu} we
372c401
< employ a free electron valence of 3 for each Al atom and 1.805 for
---
> employ a free-electron valence of 3 for each Al atom and 1.805 for
380c409
< 0.3~\AA/atom. We consider those structures to be unstable and display
---
> 0.3~\AA/atom.  We consider those structures to be unstable and display
384c413
< \alcu{3}{} in the \lsb~ structure, defining $\Delta E$.
---
> \alcu{3}{} in the \lsb~structure, defining $\Delta E$.
386c415
< Inspection of fig.~\ref{fig:Eofy} reveals that the convex hull is well
---
> Inspection of Fig.~\ref{fig:Eofy} reveals that the convex hull is well
398a428
> 
402c432
< Ternary phase diagrams for both \alconi{}{}{} and \alcocu{}{}{}
---
> Ternary phase diagrams for both Al-Co-Ni and Al-Co-Cu
405c435
< \alcocu{}{}{} (Fig.~\ref{fig:ternaries}a) exhibits several distinct
---
> Al-Co-Cu (Fig.~\ref{fig:ternaries}a) exhibits several distinct
411c441
< above in the context of binary \alcu{}{} structures. At 50\% Al
---
> above in the context of binary Al-Cu structures. At 50\% Al
423,424c453,454
< construction of successful ``mock ternary'' AlCoCu
< potentials~\cite{CW98} that started with binary AlCo
---
> construction of successful ``mock ternary'' Al-Co-Cu
> potentials~\cite{CW98} that started with binary Al-Co
432c462
< The phase diagram~\cite{Godecke} of \alconi{}{}{}
---
> The phase diagram~\cite{Godecke} of Al-Co-Ni 
434,435c464,465
< \alcocu{}{}{}. Instead of well isolated composition fields, the phase
< diagram of \alconi{}{}{} is marked by highly elongated composition
---
> Al-Co-Cu.  Instead of well isolated composition fields, the phase
> diagram of Al-Co-Ni is marked by highly elongated composition
445,448c475,478
< 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.
---
> 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.
451,452c481,482
< known, we made reasonable guesses. For example, we employed the
< Cockayne and Widom~\cite{CW98} model for the decagonal phase. A
---
> known, we made reasonable guesses.  For example, we employed the
> Cockayne and Widom~\cite{CW98} model for the decagonal phase.  A
470,471c500,501
< 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
---
> 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
474c504
< corresponding figure for binary \alcu{}{} except for the addition of
---
> corresponding figure for binary Al-Cu except for the addition of
486c516
< energies truncated at the pair potential level and neglecting
---
> energies truncated at the pair-potential level and neglecting
515,518c545,548
< 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
---
> The ability of interatomic potentials to reproduce a trend among 
> distinct compounds is a further test of their applicability. We have 
> 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
548a579
> 
565,566c596,597
< displacement of 0.14~\AA~ and mean atomic displacement of
< 0.18~\AA~ is encouraging.  However, a few Al atoms
---
> displacement of 0.14~\AA~and mean atomic displacement of
> 0.18~\AA~is encouraging.  However, a few Al atoms
568c599
< in~\cite{CW97,CW98,MarekandSuck}. We attribute these large
---
> in Refs.~\cite{CW97,CW98,MarekandSuck}. We attribute these large
584c615
< notable for AlCo and CoCo pairs, as illustrated Fig.~\ref{fig:gofr}.
---
> notable for Al-Co and Co-Co pairs, as illustrated Fig.~\ref{fig:gofr}.
586,596c617,626
< 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.
---
> broadening to mimic the effects of thermal fluctuations at $T=1000$~K.
> 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 at this separation.  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.
602c632
< 0.22~\AA~ for Al atoms, 0.15~\AA~ for Cu atoms and 0.08~\AA~ for Co
---
> 0.22~\AA~for Al atoms, 0.15~\AA~for Cu atoms and 0.08~\AA~for Co
604,605c634,635
< AlCoCu structure and highlights the strong CoCo bonds. In addition to
< the bonds drawn, there are favorable near neighbor separations
---
> Al-Co-Cu structure and highlights the strong Co-Co bonds. In addition 
> to the bonds drawn, there are favorable near-neighbor separations
613,615c643,645
< 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
---
> 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
618c648
< the pair potential~\cite{Moriarty90}. The resulting interactions
---
> the TM pair potentials~\cite{Moriarty90}. The resulting pair interactions
621c651
< of the quasicrystal phase at the pair potential level. Indeed, the
---
> of the quasicrystal phase at the pair-potential level. Indeed, the
628a659
> 
629a661
> \label{sec:conclusions}
631,633c663,667
< 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
---
> 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
636c670
< Al-Ni and AlCo, and comparisons of the ternary alloys Al-Co-Cu and
---
> Al-Ni and Al-Co, and comparisons of the ternary alloys Al-Co-Cu and
640,642c674,676
< 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
---
> 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
649c683
< Much more serious are the difficulties at larger Co concentration.
---
> More serious are the difficulties at larger Co concentrations.
651,652c685,689
< neglect of many-body interactions prohibit application of these
< potentials.
---
> 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.
675c712
< ed. M.~V. Jaric (Academic, San Diego, 1988) p. 1.
---
> ed. M.~V. Jaric (Academic, San Diego, 1988), p. 1.
684c721
< (Springer-Verlag, Berlin, 1987)
---
> (Springer-Verlag, Berlin, 1987).
687c724
< and Solids} (Clarendon Press, Oxford, 1995)
---
> and Solids} (Clarendon Press, Oxford, 1995).
690c727,728
< Ab-initio studies of Al-rich intermetallics systems include:
---
> {\it Ab-initio} electronic-structure studies of Al-rich intermetallic 
> systems include: 
701,702c739,740
< R.~F. Sabiryanov, S.~K. Bose and S.~E. Burkov, J. Phys: Cond. mat. {\bf 7}
< 5437 (1995);
---
> R.~F. Sabiryanov, S.~K. Bose and S.~E. Burkov, J. Phys: Cond. Matt. 
> {\bf 7}, 5437 (1995);
704,705c742,743
< M. Krajci, J. Hafner and M. Mihalkovic, Phys. Rev. B {\bf 56}
< 3072 (1997);
---
> M. Krajci, J. Hafner and M. Mihalkovic, Phys. Rev. B {\bf 56},
> 3072 (1997).
718c756
< (1992)
---
> (1992).
727c765
< Phys. Rev. B {\bf 49}, 8701 (1994)
---
> Phys. Rev. B {\bf 49}, 8701 (1994).
740c778
< \bibitem{MeltingMo} J.A. Moriarty, Phys. Rev. B {\bf 49}, 12431 (1994).
---
> \bibitem{MeltingMo} J.~A. Moriarty, Phys. Rev. B {\bf 49}, 12431 (1994).
757c795
< \bibitem{JLMurray} J.~L. Murray, Int. Met. Rev. {\bf 30}, 211 (1985)
---
> \bibitem{JLMurray} J.~L. Murray, Int. Met. Rev. {\bf 30}, 211 (1985).
761c799
< J. Less-Common Metals, {\bf 29} 133 (1972)
---
> J. Less-Common Metals, {\bf 29} 133 (1972).
764c802
< \bibitem{Preston}G.~D. Preston, Phil. Mag. {\bf 12} 980 (1931)
---
> \bibitem{Preston}G.~D. Preston, Phil. Mag. {\bf 12} 980 (1931).
773c811
< 705 (1989)
---
> 705 (1989).
777c815
< Ramachandrarao, Z. Metallk. {\bf 71}, 756 (1980)
---
> Ramachandrarao, Z. Metallk. {\bf 71}, 756 (1980).
782c820
< Acta Metall. {\bf 35} 727 (1987)
---
> Acta Metall. {\bf 35} 727 (1987).
787c825
< 1284 (1988)
---
> 1284 (1988).
792c830
< Z. Metallkd. {\bf 89}, 687 (1998)
---
> Z. Metallkd. {\bf 89}, 687 (1998).
796c834
< B. Grushko, Phase Trans. {\bf 44}, 99 (1993)
---
> B. Grushko, Phase Trans. {\bf 44}, 99 (1993).
800c838
< M.G. Bown and P.J. Brown, Acta Cryst. {\bf 9}, 911 (1956)
---
> M.G. Bown and P.J. Brown, Acta Cryst. {\bf 9}, 911 (1956).
804,805c842,843
< 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)
---
> 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).
808c846
< \bibitem{Raynor} G.~V. Raynor, Prog. Met. Phys. {\bf 1}, 1 (1949)
---
> \bibitem{Raynor} G.~V. Raynor, Prog. Met. Phys. {\bf 1}, 1 (1949).
812c850
< B. Grushko and K. Urban, Phil. Mag. B {\bf 70}, 1063 (1994)
---
> B. Grushko and K. Urban, Phil. Mag. B {\bf 70}, 1063 (1994).
821c859
< (World Scientific, 1995)
---
> (World Scientific, 1995).
825c863
< C. Freiburg and B. Grushko, Z. Krist. {\bf 209}, 49 (1994)
---
> C. Freiburg and B. Grushko, Z. Krist. {\bf 209}, 49 (1994).
828c866
< E. Cockayne and M. Widom, Phil. Mag. A {\bf 77}, 593 (1998)
---
> E. Cockayne and M. Widom, Phil. Mag. A {\bf 77}, 593 (1998).
835c873
< {\bf 205-207}, 701 (1996)
---
> {\bf 205-207}, 701 (1996).
839c877
< Mat. Sci. Eng. A, to appear (2000)
---
> Mat. Sci. Eng. A, to appear (2000).
843c881
< Y. Wang and M. Widom, in preparation (1999)
---
> Y. Wang and M. Widom, in preparation (1999).
850,851c888
< J.~A. Moriarty, Phys. Rev. B {\bf 42}, 1609 (1990) and {\bf 49}, 12 431
< (1994).
---
> J.~A. Moriarty, Phys. Rev. B {\bf 42}, 1609 (1990). 
1001c1038
< \vspace{10pt}
---
> %\vspace{10pt}
1003,1004c1040,1041
< Refs.~\cite{Gru93} and~\cite{Gru94}) and (b) Al-Co-Ni (adapted from
< Ref.~\cite{Godecke}).}
---
> Refs.~\protect\cite{Gru93} and~\protect\cite{Gru94}) and (b) Al-Co-Ni 
> (adapted from Ref.~\protect\cite{Godecke}).}
1013,1014c1050,1051
< $x,y \rightarrow 0$. (a) Aluminum interactions. (b) Transition metal
< interactions}
---
> $x,y \rightarrow 0$. (a) Aluminum interactions. (b) Transition-metal
> interactions.}
1021c1058
< \caption{Scatter plot of binary \alcu{1-y}{y} structural energies $E(y)$.
---
> \caption{Scatter plot of binary \alcu{1-y}{y} structural energies $\Delta E(y)$.
1027,1028c1064,1065
< unrelaxed unstable structure ($\square$). The solid line indicates the
< convex hull.}
---
> unrelaxed unstable structure ($\square$). 
> The solid line indicates the convex hull.}
1037,1038c1074,1075
< considered. Solid lines border facets of convex hull in calculated
< $E(x,y)$.}
---
> considered. Dashed lines border facets of convex hull in calculated
> $\Delta E(x,y)$.}
1045,1046c1082,1083
< \caption{Scatter plot of ternary structural energies $E(x,y)$ along the line
< $x=y$. Plotting symbols as in Fig.~\ref{fig:Eofy}}
---
> \caption{Scatter plot of ternary structural energies $\Delta E(x,y)$ 
> along the line $x=y$. Plotting symbols as in Fig.~\ref{fig:Eofy}.}
1053,1054c1090,1091
< \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}}
---
> \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}.}
1061,1062c1098,1099
< \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}}
---
> \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}.}
1074c1111
< \caption{Structure of the AlCoCu decagonal phase. Open circles indicate
---
> \caption{Structure of the Al-Co-Cu decagonal phase. Open circles indicate
1076c1113
< small atoms occur in parallel layers 2~\AA~ apart. Gray and Black
---
> small atoms occur in parallel layers 2~\AA~apart. Gray and black