17c17
< \newcommand{\dosb}[1]{\mbox{D0$_{#1}$}}
---
> \newcommand{\dzerosb}[1]{\mbox{D0$_{#1}$}}
82c82
< among atoms within these compounds should deepen our understanding
---
> among atoms within intermetallic compounds should deepen our understanding
95c95
< dependent, and the presence of transition-metal (TM) components may 
---
> dependent. The presence of transition-metal (TM) components may 
151c151
< structures or more complex structures observed in chemically similar 
---
> structures, or more complex structures observed in chemically similar 
263c263
< phases diagrams.
---
> phase diagrams.
276,290c276,292
< 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.
---
> 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.
321c323
< on the convex hull, and that \alco{13}{4} preempts the \dosb{11} 
---
> on the convex hull, and that \alco{13}{4} preempts the \dzerosb{11} 
329c331
< Since Cu is a noble metal with a completely filled $d$ shell in the
---
> Since Cu is a noble metal, with a completely filled $d$ shell in the
423c425
< stabilize this structure. Both \alco{9}{2} and \dosb{11} (the
---
> stabilize this structure. Both \alco{9}{2} and \dzerosb{11} (the
471c473
< compound \alcocu{1-x-y}{x}{y}. Figure~\ref{fig:alcocu} reproduces the
---
> compound Al-Co-Cu. Figure~\ref{fig:alcocu} reproduces the
482,496c484,500
< 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.
---
> 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.
525c529,530
< discussion and reflects the need to include many-body interactions.
---
> discussion in Sec.~\ref{sec:totE} and reflects the need to include
> many-body interactions.
528c533,534
< energy versus composition along special lines.  As noted above,
---
> energy versus composition along special lines in
> Figs.~\ref{fig:Eofx=y}-~\ref{fig:Eofx+y=1/3}.  As noted above,
530,538c536,546
< 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
---
> 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
545,550c553,558
< 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
< understand the differing solubilities of one transition metal for the
< other.
---
> 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.
554,557c562,567
< 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 $x+y=0.1818$.
---
> 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 $x+y=0.1818$.
565,566c575,576
< Still, the trend of greater Ni solubility than Cu in
< \alco{9}{2} is faithfully reproduced.
---
> The trend of greater Ni solubility than Cu in \alco{9}{2} is
> faithfully reproduced.
569c579
< \dosb{11} structure of \alni{3}{}. We find that Co can fully replace
---
> \dzerosb{11} structure of \alni{3}{}. We find that Co can fully replace
573c583
< same \dosb{11} structure is not stable for \alcu{3}{} nor is it stable
---
> same \dzerosb{11} structure is not stable for \alcu{3}{} nor is it stable
583,592c593,602
< 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.
---
> 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.
604c614
< below the tie-line with $\tau'$, M-(Al,Cu)$_{13}$Co$_4$ and
---
> below the tie-line with $\tau_3$, M-(Al,Cu)$_{13}$Co$_4$ and
617a628,630
> Without broadening the correlation functions would be a dense collection
> of closely-spaced $\delta$-functions.
> 
624c637
< no Co-Co pairs at this separation.  However, the strong second and 
---
> no Co-Co pairs with separations below 4~\AA.  However, the strong second and 
634,639c647,650
< Al-Co-Cu structure and highlights the strong Co-Co 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
---
> 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
641c652,654
< quasicrystals and approximants.
---
> 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.
654c667
< Cockayne and Widom decagonal phase model~\cite{CW98} and anneals the
---
> Cockayne and Widom decagonal phase model and anneals the
657c670
< Cockayne and Widom~\cite{CW98} based upon mock ternary potentials.
---
> Cockayne and Widom based upon mock ternary potentials.
665,672c678,687
< 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. 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.
---
> 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.
757a773,776
> %AlCuLi pair potentials
> \bibitem{Windisch94pot} M. Windisch, J. Hafner, M. Krajci and M. Mihalkovic,
> Phys. Rev. B {\bf 49}, 8701 (1994).
> 
763,766d781
< %AlCuLi pair potentials
< \bibitem{Windisch94pot} M. Windisch, J. Hafner, M. Krajci and M. Mihalkovic,
< Phys. Rev. B {\bf 49}, 8701 (1994).
< 
868c883
< \bibitem{CW98} E. Cockayne and M. Widom, Phys. Rev. Lett.
---
> \bibitem{CW98} E. Cockayne and M. Widom, Phys. Rev. Lett. {\bf 81}, 598 (1998)
883,885d897
< \bibitem{ThankMarek} We are indebted to Marek Mihalkovic for developing
< the computer program used in this simulation.
< 
889a902,904
> \bibitem{ThankMarek} We are indebted to Marek Mihalkovic for developing
> the computer program used in this simulation.
> 
904a920,924
> %Study of X-AlCoNi
> \bibitem{X}
> T. G\"{o}decke, M. Scheffer, R. L\"{u}ck, S. Ritsch and C. Beeli,
> Z. Metallkd. {\bf 88}, 687 (1997).
> 
937c957
< Fe$_3$Al    & cF16 (\dosb{3})  & Fm3m    & \cite{Pearson} & 0.2500 \\
---
> Fe$_3$Al    & cF16 (\dzerosb{3})  & Fm3m    & \cite{Pearson} & 0.2500 \\
939c959
< Al$_3$Ti    & tI8 (\dosb{22})  & I4/mmm  & \cite{Pearson} & 0.2500 \\
---
> Al$_3$Ti    & tI8 (\dzerosb{22})  & I4/mmm  & \cite{Pearson} & 0.2500 \\
941c961
< \alni{3}{}  & oP16 (\dosb{11}) & Pnma    & \cite{Pearson} & 0.2500 \\
---
> \alni{3}{}  & oP16 (\dzerosb{11}) & Pnma    & \cite{Pearson} & 0.2500 \\
980c1000
< \caption{Structural data for real and hypothetical Al-Co-Cu
---
> \caption{Structural data for hypothetical Al-Co-Cu
1040,1041c1060,1064
< Refs.~\protect\cite{Gru93} and~\protect\cite{Gru94}) and (b) Al-Co-Ni 
< (adapted from Ref.~\protect\cite{Godecke}).}
---
> 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})}