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\begin{document}

\preprint{APS/123-QED}

\title{Quasicrystals approximants with novel compositions and structures}


\author{M. Mihalkovic}
 \altaffiliation[Also at ]{Slovakian Address}
\author{M. Widom}
\affiliation{
Department of Physics, Carnegie Mellon University\\
Pittsburgh, PA  15213
}

\date{\today}

\begin{abstract}
We identify several new quasicrystal approximants in alloy systems in
which quasicrystals have not been previously reported.  One large
group of approximants occur in alloys of Boron with transition metals.
All these new approximants are structurally quite novel as well.
While all exhibit local motifs of 5-fold symmetry, suggesting the
quasicrystals should have 10-fold symmetry, it appears that in certain
cases the quasicrystal might actually prefer 12-fold symmetry.
SIMPLE MODELS WITH RIGID DECORATION RULES.
\end{abstract}


\pacs{Valid PACS appear here}
\maketitle

Axial quasicrystals are structures that possess one axis of
crystallographically forbidden rotational symmetry.  The structure
is quasiperiodic in the plane perpendicular to this axis, but can
be periodic in the direction parallel to the axis.  Since they never
exactly repeat, their lattice constant can be thought of as infinite
within the quasiperiodic plane.  Along the periodic direction the
lattice constant is finite and might be fairly small.

Quasicrystal approximants are ordinary, though complex, crystals whose
local structural motifs naturally extend to quasiperiodic structures
with the crystallographically forbidden rotational
symmetries~\cite{QCApprox}.  Approximants reproduce within their unit
cell a portion of an aperiodic structure, certain special fragments of
which can be extended periodically while maintaining reasonable local
atomic structures.  Generally approximants have large lattice
parameters in the directions in which the corresponding quasicrystal
is quasiperiodic.
  
Structures that possess at least one large lattice parameter are
thus candidates for being a quasicrystal approximant.  If, in
addition they possess a short lattice parameter orthogonal to their
large one(s) this direction is a candidate for being an
axis of high rotational symmetry in an axial quasicrystal.

Experimentally observed axial quasicrystal symmetries are octagonal
(8x), decagonal (10x) and dodecagonal (12x).  To-date decagonal
quasicrystals are known in the compounds AlCoNi, AlCoCu, AlMnPd,
AlCrNi, AlNiRu, AlCuRh, AlFeNi, ZnMgDy, AlCrFe, GaFeCuSi, AlCuCr and
possibly others.  In all these cases, the structures are believed to
be layered, with a basic 4~\AA~ unit consisting of two atomic layers
that are either stacked periodically or with some modulation leading
to periodicities of 4, 8, 12 and 16~\AA~.  Perpendicular to the axis
the structures are quasiperiodic.  Some observed metastable dodecagonal
quasicrystals occur in Ta-Te, Ni-Cr, Bi-Mn, Ni-V.

In the process of searching for bulk metallic glass-forming
compounds~\cite{alcacu_unp,bfezry_unp} we developed a database of
intermetallic structures which now contains about 1000 structures
drawn mainly from standard
references~\cite{TernaryPD,Pearson,BinaryPD,PD_new}.  The data base
contains the Pearson symbol of each structure (e.g. B$_4$Mg$_2$Ru$_5$
has Pearson symbol oP22 indicating an orthorhombic Primitive cell
with 22 atoms), the space group (e.g. number 55, or Pbam), as well as the
lattice parameters and Wyckoff coordinates.

We screened this database for structures that contain one short
lattice parameter of 5~\AA~ or less and one long lattice parameter of
8~\AA or more.  Owing to the presence of at least one long axis, these
structures tend to have relatively large numbers of atoms per unit
cell, as quoted in their Pearson symbols. We drew pictures of the
structures projected along their short axis, then visually examined
the picture.  In many cases we observed the presence of local
structural motifs of approximate 5-fold symmetry.  We took special
note of compounds that were chemically similar to the glass-forming
compounds of interest (Al-Ca-Cu and B-Fe-Y-Zr).

When we found local 5-fold symmetry we attempted to formulate a tile
decoration model that would reproduce the observed structure and allow
us to extend the structure to larger approximants or a complete
quasicrystalline structure.  We were able to do this in several cases.

Given a proposed quasicrystal model, we carried out total energy
calculations using PAW potentials~\cite{PAW} in the generalized gradient
approximation~\cite{PW91} as implemented by the program
VASP~\cite{VASP,VASP2}.  Total energies of compounds of differing
composition can be compared by measuring all energies relative to the
tie-line connecting cohesive energies of pure elements.  All
structural energies were for fully relaxed structures and were
converged to a precision of 1 meV/atom.  Because we relaxed the atomic
volume, these energy differences are the structural enthalpy of
formation, $\Delta H_{for}$, evaluated at temperature T=0K.

One interesting case is B-Mg-Ru alloys, where B$_4$Mg$_2$Ru$_5$.oP22
($10.0\times 8.4\times 3.0$~\AA$^3$) and B$_{11}$Mg$_5$Ru$_{13}$.oP62
($21.9\times 10.0\times 2.9$~\AA$^3$) are two observed
structures~\cite{oP22oP62} that we believe are approximants of a
B$_{38}$Mg$_{17}$Ru${45}$ decagonal quasicrystal.
Figure~\ref{fig:bmgru} shows the structure of oP22 and our proposed
extension to a decagonal quasicrystal based on boat-hexagon tilings.

\begin{figure}
\includegraphics[width=3 in]{B4Mg2Ru5_oP22}
\includegraphics[width=3 in]{tiling_brumg}
\caption{\label{fig:bmgru}(a)B$_4$Mg$_2$Ru$_5$ approximant. (b)
Proposed model quasicrystal. Large and small circles indicate upper
and lower layers.  Black, gray and white indicate large, medium and
small atoms, respectively.}
\end{figure}

This compound is chemically unlike any presently known for
quasicrystal formation.  Notably, it contains a fairly large
concentration of Boron.  Owing to the prevalence of icosahedra in
elemental Boron, researchers have long suggested that icosahedral
quasicrystals might occur in Boron-rich
compounds~\cite{Favio,Takeda,Weygand,Kimura,ZH,Boustani}, but none has
yet been found.

Our proposed structure is promising because we have investigated the
enthalpy of formation of other, larger hypothetical approximants.  We
consider how far these approximants lie from the convex hull of
enthalpy versus composition for all known B-Mg-Ru structures, and find
a large set of structures, nearly degenerate in energy, lying slightly
above the convex hull.  This is just hte scenario envisioned for the
entropic stabilization of quasicrystals, so we suggest that perhaps
this compound will exhibit a thermodynamically stable high-temperature
decagonal quasicrystal phase.  We will describe the tiling energetics,
including an evaluation of the energy of vertical phason stacking
faults, in an upcoming puglication~\cite{MW_unp}.

Comment on novelty of structure.

Likewise, Al$_8$CaCo$_2$.oP44 ($14.5\times 12.5\times 4.0$~\AA$^3$) is
an approximant of a hypothetical Al$_x$Ca$_y$Co$_z$ decagonal phase.
The structure here is quite similar to B-Mg-Ru, with the substitutions
Mg->Ca, b->Co exactly reproducing the B-Mg layer as a Co-Ca layer.
However, the Al-content in the adjoining atomic layer is about
3$\times$ the Ru content of the adjoining layer in the B-based
compound.  Unlike the case of B-Mg-Ru, where the star tile is
energetically unfavorable, leading to tilings of hexagons and boats
only, for Al-Ca-Co the star tile is comparable in energy to the boat
and hexagon tiles, leading to full HBS tilings.

\begin{figure}
\includegraphics[width=3 in]{Al8CaCo2_oP44}
\caption{\label{fig:brey}B-Re-Y approximant}
\end{figure}

Due to the interest in finding genuinely B-rich quasicrystals, we
screened our database to select high B concentration and found several
candidate approximants.  Noteworthy cases include B$_6$ReY$_2$.oP36,
B$_7$ReY$_3$.oC44 and B$_4$CrY.oP24.  All of these feature planar
Boron pentagons, centered by a medium-sized transition metal.  In the
first two cases, a second pentagonal ring of Y-atoms surrounds the
pentagonal ring of B-atoms.  Fig.~\ref{fig:brey} illustrates one of
the B-Re-Y structures.  We have not yet identified the proper manner
to extend these structures to a full decagonal quasicrystal in a
manner that achieves sufficiently low energy, but since they are all
stable as BFeY alloys, and occur close-by in composition, we suspect
there may be a unifying quasicrystal model to which they are
approximants.

\begin{figure}
\includegraphics[width=3 in]{B4CrY_oP24}
\caption{\label{fig:brey}B-Re-Y approximant}
\end{figure}

Another curious structure is B$_2$C$_2$Sc.oP20, which is isostructural
with B$_4$CrY.oP24 except that the pentagonal rings contain no
medium-sized transition metal. In this compound, C and B share the
pentagonal rings.

Another case that we consider promising for the occurence of an
entropically stabilized decagonal quasicrystal is Eu-Cu and the
isostructural compound Ca-Cu.  Here again we find two experimentally
observed approximants, CuEu.oP8 ($8.0\times 4.4\times 6.0$~\AA$^3$)
and CuEu2.oP12 ($6.4\times 4.2\times 15.1$~\AA$^3$) that allow us to
devise decoration rules and extend the motif to a full decagonal
quasicrystal, as shown in Fig.~\ref{fig:cueu}.  Surprisingly, one of
our larger approximant structures that we produced by hand,
Cu$_4$Eu$_9$, turns out also to be precited stable, though it has not
been reported experimentally.  The remaining approximants we tested
lie slightly above the convex hull, suggesting again the likelihood of
entropic stabilization.

\begin{figure}
\includegraphics[width=3 in]{CuEu2_oP12}
\includegraphics[width=3 in]{tiling_cueu}
\caption{\label{fig:cueu}(a) CuEu$_2$ approximant. (b) Cu$_4$Eu$_9$
approximant that we predict to be stable}
\end{figure}

As in the case of the B-Mg-Ru system, this new alloy system is
chemicaly unlike previously known quasicrystals, this time because it
is rich in a Rare Earth element. MAREK, DID YOU LOOK AT OTHER RARE
EARTHS? IS IT POSSIBLE THERE COULD BE INTERESTING MAGNETIC ORDER?
LET'S THINK ABOUT THIS LATER ON (AFTER THE MRS MEETING).

Unlike the previous structures, where the basic tiles were drawn from
HBS tilings, for Cu-Eu the basic structure is based on a decoration of
the rectangle-triangle (RT) tiling.  The rectangle-triangle tiling is
a 10$\times$ symmetric variant of the 12$\times$ symmetric
square-triangle tiling which form dodecagonal quasicrystals.
Interestingly, both these models have exact solutions for their
entropy and phason elastic constants.  They also share the property
that their phason degrees of freedom in general are quite complex
involving non-local ``zipper'' updates.

The next class of structures we mention are based on Frank-Kasper
structures.  These are tetrahedrally close-packed structures.  The
constraints of close packing dictate that only special atomic
environments occur.  The basic atomic environment is the icosahedron,
in which three atoms on each triangular face for a tetrahedron with
the central atom.  Each bond from center to shell is surrounded by
five other atoms.  Then there are the characteristic ``defect''
structures, in which 6 atoms surround a bond.  This leads to
coordination number 14 clusters, in which two defect bonds meet at the
center, coordination 15 with three defect bonds and coordination 16
with four defect bonds.

\begin{figure}
\includegraphics[width=3 in]{Co8Mn9Si3_oP74}
\includegraphics[width=3 in]{Mn4Si_oI168}
\includegraphics[width=3 in]{FeCr_tP30}
\caption{\label{fig:cueu}(a) CuEu$_2$ approximant. (b) Proposed
model quasicrystal. Replace FeCr?}
\end{figure}

MAREK TO WRITE

DISTINCTION BETWEEN RECTANGLE AND SQUARE (4$_2$ SCREW AXIS).

Mn$_4$Si extends to covering cluster.

Z12 dominance -> decagonal
Z14 dominance -> dodecagonal

Mg4Zn7 clearly binary in sizes

\begin{acknowledgments}
We wish to acknowledge useful discussions with C.L. Henley. This work
was supported in part by NSF grant DMR-0111198 and benefited from our
metallic glass research funded by DARPA. We thank the Pittsburgh
Supercomputer Center for computer time.
\end{acknowledgments}

\bibliography{decapp}

\end{document}
